(* Mathematica script 2danQCDcouplrt0201al01185.m *)

(* Copyright: G. Cvetic (November 2018), Univ. Tecnica Federico Santa Maria
 Permission is granted for anyone to use this script for any purpose 
 on any computer. The script can be freely distributed. *)

(* The script has been checked to work in Mathematica 11.3 *)
(* ************************************************************ *)

Print["2danQCDcouplrt0201al01185: a Mathematica package for analytic running"]
Print["of the strong coupling A (the analog of pQCD coupling a=alpha_s/Pi)"]
Print["in a model with 2 deltas in the spectral function of A,"]
Print["in the c2=-4.9 Lambert renormalization scheme."]
Print["This model:"]
Print["1.) agrees with perturbative QCD at high Q2 up to (Lambda^2/Q2)^5;"]
Print["2.) reproduces correctly the tau semihadronic decay physics."]
Print["The input parameters are such that r_{tau}^{(D=0)}=0.201 ('rt0201')"]
Print["in the (leading-beta0) + (beyond-leading-beta0) resummation (rtauLBbLB)"]
Print["and alpha_s(MZ^2;MSbar)=0.1185 ('al01185')."]
Print["For additional details, see arXiv:1408.6868"]
Print["The software has been checked to work in: Mathematica 11.3"]
Print["The software can be freely distributed."];
Print["Copyright: G.Cvetic, September 2018, Univ. Tecnica Federico Santa Maria"]

(* ************************************************************ *)
(* ************************************************************ *)


(* ::Section:: *)
(*Definitions*)

(* Couplings, functions: *)
LL2::usage= "LL2[Nf]=Lambda_L^2, this is square the Lambert Lambda scale Lambda_L (with Nf flavors), in GeV^2, appearing in the pQCD running coupling a(Q^2) (see: a2d) underlying for the considered Nf-flavor 2danQCD";
alphaMSMZ::usage= "alphaMSMZ=alpha_s(Q^2=MZ^2,MSbar). In this case, alphaMSMZ=0.1185"
c22din::usage= "c22din=c2=beta_2/beta_0 (chosen to be c2ddin=-4.9) is the value of the c2 coefficient (for Nf=3,4,5,6) in the Lambert renormalization scheme used in the considered 2danQCD [c2=-4.9; c3=c2^2/c1. c4=c2^3/c1^2, etc., and cj=beta_j/beta_0; normalization convention: beta_0=(11 -2*Nf/3)/4; beta_1=(102-38*Nf/3)/16; etc.]"
s0::usage= "s0[Nf]=s_0 is a dimensionless parameter for Nf flavor 2danQCD (Nf=3,4,5,6); it is equal for Nf=3,4,5,6; the pQCD-onset scale M_0^2 in Nf=3 2danQCD is then: M_0^2(Nf=3)=s0[3]*LL2[3] (in GeV^2), and similarly for other cases of Nf"
s1s0::usage= "s1s0[Nf]=s_1 is a dimensionless parameter for Nf flavor 2danQCD (Nf=3,4,5,6); the location of the first delta function of the discontinuity function rho_1(sigma)=Im A(Q^2=-sigma - I epsilon) of the A-coupling of the Nf-flavor 2danQCD is then at sigma=M_1^2=s1s0[Nf]*LL2[Nf] (in GeV^2)"
s2s0::usage= "s2s0[Nf]=s_2 is a dimensionless parameter for Nf flavor 2danQCD (Nf=3,4,5,6); the location of the second delta function of the discontinuity function rho_1(sigma)=Im A(Q^2=-sigma - I epsilon) of the A-coupling of the Nf-flavor 2danQCD is then at sigma=M_2^2=s2s0[Nf]*LL2[Nf] (in GeV^2)"
f1ss0::usage= "f1ss0[Nf]=f_1^2 is a dimensionless strength parameter at the first delta in rho_1(sigma) discontinuity function of A(Q^2), in the Nf-flavor 2danQCD; the first delta function in rho_1(sigma) is Pi*f1ss0[Nf]*LL2[Nf]*delta(sigma - M1^2[Nf]), where M1^2[Nf]= s1s0[Mf]*LL2[Nf] (in GeV^2)"
f2ss0::usage= "f2ss0[Nf]=f_2^2 is a dimensionless strength parameter at the second delta in rho_1(sigma) discontinuity function of A(Q^2), in the Nf-flavor 2danQCD; the second delta function in rho_1(sigma) is Pi*f2ss0[Nf]*LL2[Nf]*delta(sigma - M2^2[Nf]), where M2^2[Nf]= s2s0[Mf]*LL2[Nf] (in GeV^2)"  
a2d::usage= "a2d[Nf,u,fi]= alpha_s(Q^2=u*LL2[Nf]*Exp[I fi)/Pi =a(Q^2), where Q^2=|Q^2|*Exp[I fi] and |Q^2|=u*LL2[Nf] (see: LL2); a2d is the underlying pQCD coupling in the Lambert renormalization scheme [c2=c22din=-4.9; c3=c2^2/c1. c4=c2^3/c1^2, etc., and cj=beta_j/beta_0; normalization conventions: beta_0=(11 -2*Nf/3)/4; beta_1=(102-38*Nf/3)/16; etc.], the pQCD coupling underlying the Nf-flavor 2danQCD coupling A (see: A1).";
apert::usage= "apert[Nf,Q2,fi]= alpha_s(Q^2=Q2*Exp[I fi)/Pi =a(Q^2), where Q^2=|Q^2|*Exp[I fi] and |Q^2|=Q2; apert is the underlying pQCD coupling in the Lambert renormalization scheme [c2=c22din=-4.9; c3=c2^2/c1. c4=c2^3/c1^2, etc., and cj=beta_j/beta_0; normalization conventions: beta_0=(11 -2*Nf/3)/4; beta_1=(102-38*Nf/3)/16; etc.], the pQCD coupling underlying the Nf-flavor 2danQCD coupling A (see: A1); note, apert[Nf,Q2,fi]=a2d[Nf,Q2/LL2[Nf],fi].";
tr12d::usage= "tr12d[Nf,s] is the discontinuity function rho_1^{(pt)}(sigma)=Im a(Q^2=-sigma - I epsilon) of the underlying pQCD coupling a(Q^2) (see: a2d and apert) in the considered Nf-flavor 2danQCD, where s=sigma/LL2[Nf] (see: LL2)"
A1::usage= "A1[Nf,Q2,fi] = A(Q^2) = A_1(Q^2), where Q^2=Q2 Exp[I fi] (Q2=|Q^2|, in GeV^2) and A(Q^2) is the 2danQCD analog of the underlying perturbative coupling a(Q^2) = a2d[Nf,Q2/LL2[Nf],fi] (see: a2d); at high |Q^2|> 1 GeV^2 the two couplings practically coincide";
tA2::usage= "tA2[Nf,Q2,fi]={tilde A}_2(Q^2) for Q^2=Q2 Exp[I fi] (Q2=|Q^2|, in GeV^2); this is the first logarithmic derivative of A_1 (see: A1): {tilde A}_2(Q^2) = (-1)/(beta_0 1!) (d/d ln Q^2) A_1(Q^2)";
tA3::usage= "tA3[Nf,Q2,fi]={tilde A}_3(Q^2) for Q^2=Q2 Exp[I fi]; this is the second logarithmic derivative of A_1 (see: A1): {tilde A}_3(Q^2) = (-1)^2/(beta_0^2 2!) (d/d ln Q^2)^2 A_1(Q^2)";
tA4::usage= "tA4[Nf,Q2,fi]={tilde A}_4(Q^2) for Q^2=Q2 Exp[I fi]; this is the third logarithmic derivative of A_1 (see: A1): {tilde A}_4(Q^2) = (-1)^3/(beta_0^3 3!) (d/d ln Q^2)^3 A_1(Q^2)";
tA2d::usage= "tA2d[Nf,nu,Q2,fi]= {tilde A}_{nu}(Q^2), with Q^2=Q2 Exp[I fi]  (Q2=|Q^2|); {tilde A}_{nu} can have nu noninteger; {tilde A}_{nu} is the generalization (to noninteger nu) of the logarithmic derivative (-1)^{nu-1}/(beta_0^{nu-1} (nu-1)! (d/d ln Q^2)^(nu-1) A_1(Q^2)";
A2dNX::usage= "A2dNX[Nf,M,nu,Q2,fi] = A_{nu+M}(Q^2) is the 2danQCD analog of the power of the pQCD coupling a2d[Nf,Q2/LL2[Nf],fi]^(nu+M)=a(Q^2)^(nu+M) (see: a2d), for NX=N1,N2,N3,N4; here, -1 =< nu =< 1 (in general noninteger), and M is integer, M=0,1,..,X; where Q^2=Q2 Exp[I fi] and Q2=|Q^2|=u*LL2[nq] (in GeV^2). Note that for X=1 (N1), A2dN1[...] are the couplings (A_{nu}, A_{nu+1}) in the NLO sense, when in the TPS of a considered observable we only have two terms known (~a^nu and ~a^(nu+1)); for X=2 (N2), A2dN2[...] are the couplings (A_{nu}, A_{nu+1}, A_{nu+2}) in the N^2LO sense, i.e.,  when in the truncated perturbation series of a considered observable we have three terms known (~a^nu, ~a^(nu+1) and ~a^(nu+2)); etc.; Nf indicates the chosen number of active flavors of the 2danQCD (Nf=3,4,5,6), usually Nf=3."
AYNX::usage= "AYNX[Nf,Q2,fi] = A_{Y}(Q^2) is the 2danQCD analog of the power of the pQCD coupling a2d[Nf,Q2/LL2[Nf],fi]^Y=a(Q^2)^Y, for NX=N1,N2,N3,N4; here, Y is integer, Y=1,..,X+1; where Q^2=Q2*Exp[I fi] and Q2=|Q^2|= u*LL2[nq] (in GeV^2). Note that X=1 (N1) is used when we have in the truncated perturbation series of a considered observable only two terms (~a and ~a^2) known; X=2 (N2) is used when three terms are known (~a, ~a^2 and ~a^3); X=3 (N3) is used when four terms are known (~a, ~a^2, ~a^3 and ~a^4); X=4 (N4) is used when five terms are known (~a, ~a^2, ~a^3, ~a^4, and ~a^5). Nf indicates the chosen number of active flavors of the 2danQCD (Nf=3,4,5,6), usually Nf=3."
tdj::usage= "tdj={tilde d}_j (j=1,2,3) are the coefficients for the Nf=3 canonical (and massless) Adler function in the Lambert renormalization scheme used in the considered 2danQCD and in the underlying pQCD [c2=-4.9; c3=c2^2/c1. c4=c2^3/c1^2, etc., and cj=beta_j/beta_0; normalizations: beta_0=(11 -2 Nf/3)/4; beta_1=(102-38 Nf/3)/16; etc.]"
tdjbLB::usage= "tdjbLB={tilde d}_j - {tilde d}_j^(LB) are the 'beyond-the-leading-beta0' (bLB) part of {tilde d}_j Nf=3 coefficients of the canonical (and massless) Adler function in the Lambert renormalization scheme used in the considered 2danQCD and in the underlying pQCD (see also: tdj)."
rtauLBbLB::usage= "rtauLBbLB=r_{tau}(D=0) calculated (in the considered Lambert renormalization scheme) in the considered 2danQCD with the 'leading-beta_0 (LB) + beyond-the leading-beta_0 (bLB)' approach with Nf=3. Note that here rtauLBbLB=0.2010"
rtaudBG22::usage= "rtaudBG22=r_{tau}(D=0) calculated in the considered 2danQCD with the diagonal [2/2] Baker-Gammel approach."
rtauTPS::usage= "rtauTPS=r_{tau}(D=0) calculated in the considered 2danQCD with the truncated perturbation series approach."
aMSnf3::usage= "aMSnf3[Q2] is the pQCD coupling a(Q^2=Q2)=alpha_s(Q2)/Pi for positive Q2 (Q2 in GeV^2) in MSbar renormalization scheme, for the number of active flavors Nf=3; it corresponds to alpha_s(MZ^2;MSbar)=0.1185, and is obtained by 4-loop RGE evolution in MSbar scheme from Q2=M_Z^2 downwards, with 3-loop quark threshold conditions imposed at threshold momenta Q2=(2*mqbar)^2, where mqbar=mqbar(Q2=mqbar^2) are the MSbar quark masses of b (for Nf=5->4) and of c (for Nf=4->3); we used mbbar=4.2 GeV and mcbar=1.27 GeV."

Off[NIntegrate::inumr]

s0[3] = 25.6
 
s0[4] = 25.6
 
s0[5] = 25.6
 
s0[6] = 25.6
 
s0 /: s0::usage = "s0[Nf]=s_0 is a dimensionless parameter for Nf flavor \
2danQCD (Nf=3,4,5,6); it is equal for Nf=3,4,5,6; the pQCD-onset scale M_0^2 \
in Nf=3 2danQCD is then: M_0^2(Nf=3)=s0[3]*LL2[3] (in GeV^2), and similarly \
for other cases of Nf"
 
s1s0[3] = 18.72691887743575
 
s1s0[4] = 18.549796979734467
 
s1s0[5] = 18.20998738271616
 
s1s0[6] = 17.348971891256497
 
s1s0 /: s1s0::usage = "s1s0[Nf]=s_1 is a dimensionless parameter for Nf \
flavor 2danQCD (Nf=3,4,5,6); the location of the first delta function of the \
discontinuity function rho_1(sigma)=Im A(Q^2=-sigma - I epsilon) of the \
A-coupling of the Nf-flavor 2danQCD is then at sigma=M_1^2=s1s0[Nf]*LL2[Nf] \
(in GeV^2)"
 
s2s0[3] = 1.0350711959987655
 
s2s0[4] = 0.3049322781426228
 
s2s0[5] = -1.298264539528074
 
s2s0[6] = -7.227156864726678
 
s2s0 /: s2s0::usage = "s2s0[Nf]=s_2 is a dimensionless parameter for Nf \
flavor 2danQCD (Nf=3,4,5,6); the location of the second delta function of the \
discontinuity function rho_1(sigma)=Im A(Q^2=-sigma - I epsilon) of the \
A-coupling of the Nf-flavor 2danQCD is then at sigma=M_2^2=s2s0[Nf]*LL2[Nf] \
(in GeV^2)"
 
f1ss0[3] = 0.29283439434956976
 
f1ss0[4] = 0.3766834220334265
 
f1ss0[5] = 0.5386855942310698
 
f1ss0[6] = 0.9143907119839362
 
f1ss0 /: f1ss0::usage = "f1ss0[Nf]=f_1^2 is a dimensionless strength \
parameter at the first delta in rho_1(sigma) discontinuity function of \
A(Q^2), in the Nf-flavor 2danQCD; the first delta function in rho_1(sigma) is \
Pi*f1ss0[Nf]*LL2[Nf]*delta(sigma - M1^2[Nf]), where M1^2[Nf]= \
s1s0[Mf]*LL2[Nf] (in GeV^2)"
 
f2ss0[3] = 0.5746543918846351
 
f2ss0[4] = 0.8098093174608728
 
f2ss0[5] = 1.3937271773561326
 
f2ss0[6] = 4.071413140403119
 
f2ss0 /: f2ss0::usage = "f2ss0[Nf]=f_2^2 is a dimensionless strength \
parameter at the second delta in rho_1(sigma) discontinuity function of \
A(Q^2), in the Nf-flavor 2danQCD; the second delta function in rho_1(sigma) \
is Pi*f2ss0[Nf]*LL2[Nf]*delta(sigma - M2^2[Nf]), where M2^2[Nf]= \
s2s0[Mf]*LL2[Nf] (in GeV^2)"
 
M0[3] = 1.2974143813909098
 
M0[4] = 0.9229366765505262
 
M0[5] = 0.487833660050335
 
M0[6] = 0.11840201937085226
 
LLamb[3] = 0.25642403214085324
 
LLamb[4] = 0.18241137712536115
 
LLamb[5] = 0.09641659281595936
 
LLamb[6] = 0.023401253798454362
 
LL2[3] = 0.06575328425937332
 
LL2[4] = 0.03327391050477073
 
LL2[5] = 0.009296159370238508
 
LL2[6] = 0.0005476186793396747
 
LL2 /: LL2::usage = "LL2[Nf]=Lambda_L^2, this is square the Lambert Lambda \
scale Lambda_L (with Nf flavors), in GeV^2, appearing in the pQCD running \
coupling a(Q^2) (see: a2d) underlying for the considered Nf-flavor 2danQCD"
 
be0[nq_] := (11 - 2*(nq/3))/4
 
c1[nq_] := (102 - 38*(nq/3))/(11 - 2*(nq/3))/4
 
c22d[nq_] := c22din
 
c22din = -4.9
 
c22din /: c22din::usage = "c22din=c2=beta_2/beta_0 (chosen to be c2ddin=-4.9) \
is the value of the c2 coefficient (for Nf=3,4,5,6) in the Lambert \
renormalization scheme used in the considered 2danQCD [c2=-4.9; c3=c2^2/c1. \
c4=c2^3/c1^2, etc., and cj=beta_j/beta_0; normalization convention: \
beta_0=(11 -2*Nf/3)/4; beta_1=(102-38*Nf/3)/16; etc.]"
 
c32d[nq_] := c22d[nq]^2/c1[nq]
 
c42d[nq_] := c22d[nq]^3/c1[nq]^2
 
c52d[nq_] := c22d[nq]^4/c1[nq]^3
 
c2MS[nq_] := be2MS[nq]/be0[nq]
 
be2MS[nq_] := (2857/2 - 5033*(nq/18) + 325*(nq^2/54))/64
 
c3MS[nq_] := be3MS[nq]/be0[nq]
 
be3MS[nq_] := (1/256)*((149753/6 + 3564*Zeta[3]) - 
      (1078361/162 + (6508/27)*Zeta[3])*nq + (50065/162 + (6472/81)*Zeta[3])*
       nq^2 + (1093/729)*nq^3)
 
btP11[x_, nq_] := betaP11[x, nq]
 
betaP11[x_, nq_] := (-be0[nq])*x^2*((1 + (c1[nq] - c22d[nq]/c1[nq])*x)/
      (1 - (c22d[nq]/c1[nq])*x))
 
betaMS4l[x_, nq_] := (-be0[nq])*x^2*(1 + c1[nq]*x + c2MS[nq]*x^2 + 
      c3MS[nq]*x^3)
 
kappa2d = 2.
 
alphaMSMZ = 0.1185
 
alphaMSMZ /: alphaMSMZ::usage = "alphaMSMZ=alpha_s(Q^2=MZ^2,MSbar). In this \
case, alphaMSMZ=0.1185, or 0.1181, or 0.1189"
 
apert[nq_, Q2_, fi_] := a2d[nq, Q2/LL2[nq], fi]
 
a2d[nq_, u_, fi_] := If[fi >= 0, (-(1/c1[nq]))*
      (1/(1 - c22d[nq]/c1[nq]^2 + ProductLog[-1, zu2d[nq, u, fi]])), 
     (-(1/c1[nq]))*(1/(1 - c22d[nq]/c1[nq]^2 + ProductLog[1, 
         zu2d[nq, u, fi]]))]
 
a2d /: a2d::usage = "a2d[Nf,u,fi]= alpha_s(Q^2=u*LL2[Nf]*Exp[I fi)/Pi \
=a(Q^2), where Q^2=|Q^2|*Exp[I fi] and |Q^2|=u*LL2[Nf] (see: LL2); a2d is the \
underlying pQCD coupling in the Lambert renormalization scheme \
[c2=c22din=-4.9; c3=c2^2/c1. c4=c2^3/c1^2, etc., and cj=beta_j/beta_0; \
normalization conventions: beta_0=(11 -2*Nf/3)/4; beta_1=(102-38*Nf/3)/16; \
etc.], the pQCD coupling underlying the Nf-flavor 2danQCD coupling A (see: \
A1)."
 
zu2d[nq_, u_, fi_] := ((-(1/c1[nq]/E))*Exp[(-I)*(be0[nq]/c1[nq])*fi])/
     u^(be0[nq]/c1[nq])
 
aMSnf3[Qf2_] := First[y[Log[Qf2/Q4^2]] /. sol1a]
 
Q4 = 2.54
 
sol1a = {{y -> InterpolatingFunction[{{-2.8569647706100354, 20.}}, 
        {5, 7, 1, {656}, {4}, 0, 0, 0, 0, Automatic, {}, {}, False}, 
        {{-2.8569647706100354, -2.8569647706099928, -2.8569647706099497, 
         -2.856964770609907, -2.8569647706098644, -2.856964770609813, 
         -2.8569647706097614, -2.85696477060971, -2.8569647706096584, 
         -2.8569647706095966, -2.8569647706095345, -2.8569647706094727, 
         -2.8569647706094106, -2.856964770609336, -2.8569647706092614, 
         -2.8569647706091867, -2.856964770609112, -2.8569647706090224, 
         -2.8569647706089327, -2.856964770608843, -2.8569647706087533, 
         -2.856964770608645, -2.856964770608537, -2.8569647706084287, 
         -2.8569647706083208, -2.8569647706081907, -2.8569647706080605, 
         -2.8569647706079304, -2.8569647706078003, -2.8569647706076435, 
         -2.8569647706074868, -2.85696477060733, -2.8569647706071732, 
         -2.8569647706069845, -2.856964770606796, -2.8569647706066075, 
         -2.8569647706064187, -2.856964770606192, -2.8569647706059644, 
         -2.8569647706057375, -2.85696477060551, -2.8569647706052366, 
         -2.8569647706049635, -2.85696477060469, -2.8569647706044163, 
         -2.8569647706040873, -2.8569647706037578, -2.8569647706034287, 
         -2.856964770603099, -2.856964770602703, -2.8569647706023065, 
         -2.85696477060191, -2.8569647706015138, -2.8569647706010364, 
         -2.856964770600559, -2.8569647706000816, -2.856964770599604, 
         -2.8569647705990295, -2.856964770598455, -2.8569647705978802, 
         -2.8569647705973056, -2.8569647705966137, -2.856964770595922, 
         -2.85696477059523, -2.8569647705945376, -2.8569647705937045, 
         -2.8569647705928713, -2.8569647705920382, -2.856964770591205, 
         -2.8569647705902015, -2.8569647705891983, -2.856964770588195, 
         -2.856964770587192, -2.856964770585984, -2.856964770584776, 
         -2.8569647705835677, -2.8569647705823598, -2.8569647705809054, 
         -2.8569647705794505, -2.8569647705779957, -2.8569647705765413, 
         -2.85696477057479, -2.856964770573038, -2.8569647705712864, 
         -2.856964770569535, -2.8569647705674255, -2.856964770565316, 
         -2.856964770563207, -2.8569647705610977, -2.8569647705585575, 
         -2.8569647705560177, -2.8569647705534775, -2.8569647705509373, 
         -2.856964770547879, -2.85696477054482, -2.856964770541761, 
         -2.8569647705387022, -2.856964770535019, -2.8569647705313352, 
         -2.8569647705276515, -2.856964770523968, -2.856964770519532, 
         -2.856964770515096, -2.85696477051066, -2.8569647705062233, 
         -2.856964770500881, -2.8569647704955385, -2.8569647704901957, 
         -2.8569647704848533, -2.8569647704784193, -2.856964770471985, 
         -2.8569647704655505, -2.856964770459116, -2.856964770451367, 
         -2.856964770443618, -2.8569647704358685, -2.856964770428119, 
         -2.8569647704187857, -2.8569647704094523, -2.856964770400119, 
         -2.8569647703907854, -2.8569647703795438, -2.856964770368302, 
         -2.8569647703570604, -2.8569647703458187, -2.8569647703322785, 
         -2.8569647703187377, -2.8569647703051975, -2.856964770291657, 
         -2.856964770275347, -2.8569647702590375, -2.8569647702427274, 
         -2.856964770226418, -2.8569647702067713, -2.8569647701871252, 
         -2.8569647701674787, -2.8569647701478327, -2.8569647701241667, 
         -2.8569647701005003, -2.8569647700768344, -2.8569647700531684, 
         -2.8569647700246588, -2.8569647699961496, -2.85696476996764, 
         -2.8569647699391307, -2.8569647699047853, -2.85696476987044, 
         -2.8569647698360945, -2.856964769801749, -2.856964769760372, 
         -2.8569647697189944, -2.856964769677617, -2.8569647696362392, 
         -2.856964769586388, -2.8569647695365363, -2.8569647694866847, 
         -2.8569647694368334, -2.8569647693767695, -2.856964769316706, 
         -2.856964769256642, -2.8569647691965785, -2.856964769124207, 
         -2.856964769051836, -2.8569647689794646, -2.8569647689070936, 
         -2.8569647688198887, -2.856964768732684, -2.856964768645479, 
         -2.8569647685582744, -2.85696476845319, -2.856964768348106, 
         -2.8569647682430217, -2.8569647681379378, -2.8569647680113017, 
         -2.856964767884666, -2.85696476775803, -2.856964767631394, 
         -2.8569647674787784, -2.8569647673261622, -2.856964767173546, 
         -2.85696476702093, -2.856964766836993, -2.8569647666530558, 
         -2.856964766469119, -2.856964766285182, -2.8569647660634825, 
         -2.8569647658417834, -2.8569647656200843, -2.8569647653983847, 
         -2.8569647651311536, -2.856964764863923, -2.856964764596692, 
         -2.8569647643294607, -2.856964764007325, -2.85696476368519, 
         -2.8569647633630546, -2.856964763040919, -2.856964762652572, 
         -2.8569647622642247, -2.856964761875877, -2.85696476148753, 
         -2.8569647610193276, -2.8569647605511252, -2.8569647600829233, 
         -2.856964759614721, -2.8569647590502, -2.856964758485679, 
         -2.8569647579211583, -2.8569647573566375, -2.856964756675929, 
         -2.8569647559952207, -2.8569647553145128, -2.8569647546338044, 
         -2.856964753812927, -2.8569647529920497, -2.856964752171172, 
         -2.856964751350295, -2.856964750360299, -2.8569647493703028, 
         -2.8569647483803067, -2.8569647473903106, -2.856964746196245, 
         -2.8569647450021787, -2.856964743808113, -2.856964742614047, 
         -2.8569647411737082, -2.8569647397333697, -2.8569647382930308, 
         -2.8569647368526923, -2.856964735115114, -2.8569647333775356, 
         -2.8569647316399576, -2.8569647299023795, -2.8569647278060013, 
         -2.8569647257096236, -2.856964723613246, -2.856964721516868, 
         -2.8569647189873244, -2.8569647164577803, -2.856964713928236, 
         -2.8569647113986925, -2.8569647083461294, -2.8569647052935663, 
         -2.856964702241003, -2.85696469918844, -2.8569646955042756, 
         -2.856964691820111, -2.856964688135946, -2.8569646844517815, 
         -2.8569646800047743, -2.856964675557767, -2.85696467111076, 
         -2.856964666663753, -2.856964661295246, -2.8569646559267396, 
         -2.856964650558233, -2.856964645189726, -2.856964638707881, 
         -2.8569646322260356, -2.8569646257441903, -2.856964619262345, 
         -2.8569646114351523, -2.8569646036079592, -2.856964595780766, 
         -2.8569645879535734, -2.8569645785003814, -2.85696456904719, 
         -2.856964559593998, -2.8569645501408063, -2.8569645387220466, 
         -2.8569645273032873, -2.856964515884528, -2.8569645044657683, 
         -2.856964490670489, -2.85696447687521, -2.856964463079931, 
         -2.8569644492846518, -2.856964432615388, -2.8569644159461243, 
         -2.856964399276861, -2.856964382607597, -2.8569643624620045, 
         -2.8569643423164113, -2.8569643221708185, -2.8569643020252253, 
         -2.8569642776737694, -2.8569642533223134, -2.8569642289708574, 
         -2.8569642046194015, -2.8569641751782555, -2.856964145737109, 
         -2.856964116295963, -2.8569640868548167, -2.8569640512529126, 
         -2.8569640156510085, -2.8569639800491045, -2.8569639444472004, 
         -2.8569639013861656, -2.8569638583251313, -2.856963815264097, 
         -2.8569637722030627, -2.85696372010848, -2.856963668013897, 
         -2.8569636159193146, -2.8569635638247317, -2.856963500786811, 
         -2.8569634377488904, -2.8569633747109697, -2.8569633116730486, 
         -2.8569632353743875, -2.8569631590757263, -2.856963082777065, 
         -2.856963006478404, -2.8569629141059707, -2.856962821733538, 
         -2.856962729361105, -2.8569626369886723, -2.8569625251265203, 
         -2.8569624132643683, -2.856962301402216, -2.856962189540064, 
         -2.856962054038477, -2.8569619185368897, -2.856961783035303, 
         -2.8569616475337156, -2.856961483349529, -2.8569613191653422, 
         -2.8569611549811555, -2.856960990796969, -2.8569607917985955, 
         -2.8569605928002217, -2.8569603938018484, -2.856960194803475, 
         -2.856959953532693, -2.856959712261911, -2.856959470991129, 
         -2.856959229720347, -2.8569589371010795, -2.8569586444818116, 
         -2.856958351862544, -2.8569580592432766, -2.8569577042253327, 
         -2.8569573492073888, -2.856956994189445, -2.856956639171501, 
         -2.856956208294465, -2.8569557774174292, -2.8569553465403934, 
         -2.8569549156633576, -2.85695439252235, -2.8569538693813428, 
         -2.8569533462403354, -2.856952823099328, -2.8569521876900885, 
         -2.856951552280849, -2.856950916871609, -2.8569502814623697, 
         -2.8569495093777992, -2.856948737293229, -2.8569479652086587, 
         -2.856947193124088, -2.856946254567801, -2.8569453160115135, 
         -2.856944377455226, -2.8569434388989388, -2.8569422974731675, 
         -2.856941156047396, -2.8569400146216246, -2.856938873195853, 
         -2.856937484410865, -2.856936095625877, -2.8569347068408892, 
         -2.856933318055902, -2.8569316274953667, -2.8569299369348315, 
         -2.856928246374297, -2.8569265558137618, -2.856924496874656, 
         -2.85692243793555, -2.8569203789964446, -2.8569183200573387, 
         -2.8569158111631783, -2.8569133022690174, -2.856910793374857, 
         -2.8569082844806966, -2.856905225642121, -2.856902166803545, 
         -2.8568991079649693, -2.8568960491263935, -2.8568923176921186, 
         -2.8568885862578433, -2.8568848548235684, -2.8568811233892935, 
         -2.8568765687915776, -2.8568720141938613, -2.8568674595961454, 
         -2.8568629049984295, -2.8568573422483867, -2.8568517794983443, 
         -2.8568462167483015, -2.856840653998259, -2.8568338556250064, 
         -2.856827057251754, -2.856820258878501, -2.8568134605052484, 
         -2.856805146555619, -2.856796832605989, -2.8567885186563595, 
         -2.85678020470673, -2.856770030324927, -2.8567598559431246, 
         -2.856749681561322, -2.856739507179519, -2.8567270471631443, 
         -2.856714587146769, -2.856702127130394, -2.8566896671140194, 
         -2.856674396687491, -2.8566591262609617, -2.856643855834433, 
         -2.856628585407904, -2.856609856257499, -2.8565911271070936, 
         -2.8565723979566884, -2.856553668806283, -2.8565306791714993, 
         -2.8565076895367154, -2.8564846999019315, -2.8564617102671477, 
         -2.8564334675661818, -2.8564052248652163, -2.8563769821642504, 
         -2.8563487394632845, -2.8563140135273635, -2.856279287591442, 
         -2.8562445616555205, -2.8562098357195995, -2.8561671001974376, 
         -2.8561243646752756, -2.856081629153114, -2.856038893630952, 
         -2.8559862524695565, -2.8559336113081604, -2.8558809701467647, 
         -2.8558283289853685, -2.85576342405633, -2.855698519127292, 
         -2.8556336141982537, -2.8555687092692152, -2.855488604138365, 
         -2.855408499007514, -2.855328393876664, -2.855248288745813, 
         -2.8551493222109894, -2.8550503556761657, -2.854951389141342, 
         -2.8548524226065184, -2.8547300238654336, -2.854607625124349, 
         -2.854485226383264, -2.8543628276421793, -2.8542112826858967, 
         -2.854059737729614, -2.8539081927733316, -2.853756647817049, 
         -2.8535688037263545, -2.85338095963566, -2.8531931155449657, 
         -2.853005271454271, -2.8527721615127777, -2.8525390515712843, 
         -2.852305941629791, -2.8520728316882975, -2.851783199366392, 
         -2.851493567044486, -2.8512039347225806, -2.850914302400675, 
         -2.8505539959797375, -2.8501936895588003, -2.849833383137863, 
         -2.8494730767169254, -2.849024278185663, -2.8485754796544005, 
         -2.848126681123138, -2.8476778825918756, -2.847118123896522, 
         -2.846558365201168, -2.845998606505814, -2.84543884781046, 
         -2.844739753846921, -2.8440406598833823, -2.843341565919843, 
         -2.8426424719563044, -2.8417681519004123, -2.8408938318445203, 
         -2.840019511788628, -2.839145191732736, -2.8380501771659645, 
         -2.836955162599193, -2.835860148032421, -2.834765133465649, 
         -2.8333917316387636, -2.832018329811878, -2.830644927984992, 
         -2.8292715261581063, -2.827546417515256, -2.8258213088724053, 
         -2.8240962002295547, -2.822371091586704, -2.8202009495734055, 
         -2.8180308075601066, -2.8158606655468077, -2.8136905235335092, 
         -2.810956368118141, -2.808222212702773, -2.805488057287405, 
         -2.802753901872037, -2.799303811422149, -2.7958537209722616, 
         -2.7924036305223736, -2.788953540072486, -2.784593233149136, 
         -2.7802329262257865, -2.7758726193024366, -2.7715123123790866, 
         -2.7659929637601572, -2.7604736151412275, -2.7549542665222977, 
         -2.749434917903368, -2.742437381635493, -2.735439845367619, 
         -2.7284423090997443, -2.7214447728318696, -2.712559137057768, 
         -2.703673501283667, -2.694787865509565, -2.685902229735464, 
         -2.6746013144546787, -2.6633003991738935, -2.6519994838931082, 
         -2.640698568612323, -2.6263036542403406, -2.6119087398683583, 
         -2.5975138254963754, -2.583118911124393, -2.5647563908329047, 
         -2.546393870541417, -2.5280313502499285, -2.50966882995844, 
         -2.486225546496873, -2.4627822630353067, -2.4393389795737397, 
         -2.4158956961121727, -2.3860632516314713, -2.35623080715077, 
         -2.3263983626700684, -2.296565918189367, -2.2607175754685307, 
         -2.2248692327476944, -2.1890208900268586, -2.1433035995659626, 
         -2.097586309105067, -2.0518690186441715, -2.006151728183276, 
         -1.951124118578255, -1.8960965089732342, -1.8410688993682136, 
         -1.7707646773373855, -1.7004604553065574, -1.6301562332757291, 
         -1.559852011244901, -1.4751690028429572, -1.3904859944410133, 
         -1.3058029860390694, -1.1980740982281999, -1.0903452104173303, 
         -0.9826163226064608, -0.8748874347955914, -0.7671585469847219, 
         -0.6594296591738523, -0.5517007713629828, -0.4638656362858161, 
         -0.37603050120864945, -0.28819536613148283, -0.20036023105431616, 
         -0.1125250959771495, -0.08695121052779735, -0.061377325078445184, 
         -0.03580343962909303, -0.010229554179740865, -0.007672165634805649, 
         -0.005114777089870432, -0.002557388544935216, -0.001278694272467608, 
         0., 0.0012803175779468165, 0.002560635155893633, 
         0.005121270311787266, 0.0076819054676808985, 0.010242540623574532, 
         0.03584889218251086, 0.061455243741447195, 0.08706159530038353, 
         0.11266794685931986, 0.20616522162158532, 0.2996624963838508, 
         0.3931597711461162, 0.4866570459083817, 0.5801543206706472, 
         0.7201232229833465, 0.860092125296046, 1.0000610276087454, 
         1.1400299299214447, 1.2799988322341442, 1.4199677345468436, 
         1.6172741859267061, 1.8145806373065687, 2.0118870886864313, 
         2.209193540066294, 2.4064999914461565, 2.603806442826019, 
         2.8553484748351226, 3.1068905068442256, 3.3584325388533287, 
         3.6099745708624322, 3.8615166028715353, 4.113058634880638, 
         4.436498805828549, 4.75993897677646, 5.08337914772437, 
         5.406819318672281, 5.730259489620192, 6.053699660568102, 
         6.4706943089628135, 6.8876889573575255, 7.304683605752237, 
         7.721678254146949, 8.13867290254166, 8.555667550936372, 
         9.095449963665795, 9.63523237639522, 10.175014789124642, 
         10.714797201854067, 11.25457961458349, 11.794362027312914, 
         12.496449875182066, 13.198537723051217, 13.900625570920367, 
         14.602713418789518, 15.30480126665867, 16.006889114527823, 
         16.925226017953552, 17.843562921379284, 18.761899824805013, 
         19.380949912402507, 20.}}, {Developer`PackedArrayForm, {0, 2, 4, 6, 
         8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 
         42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 
         76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 
         108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 
         134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 
         160, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 
         186, 188, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 
         212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, 234, 236, 
         238, 240, 242, 244, 246, 248, 250, 252, 254, 256, 258, 260, 262, 
         264, 266, 268, 270, 272, 274, 276, 278, 280, 282, 284, 286, 288, 
         290, 292, 294, 296, 298, 300, 302, 304, 306, 308, 310, 312, 314, 
         316, 318, 320, 322, 324, 326, 328, 330, 332, 334, 336, 338, 340, 
         342, 344, 346, 348, 350, 352, 354, 356, 358, 360, 362, 364, 366, 
         368, 370, 372, 374, 376, 378, 380, 382, 384, 386, 388, 390, 392, 
         394, 396, 398, 400, 402, 404, 406, 408, 410, 412, 414, 416, 418, 
         420, 422, 424, 426, 428, 430, 432, 434, 436, 438, 440, 442, 444, 
         446, 448, 450, 452, 454, 456, 458, 460, 462, 464, 466, 468, 470, 
         472, 474, 476, 478, 480, 482, 484, 486, 488, 490, 492, 494, 496, 
         498, 500, 502, 504, 506, 508, 510, 512, 514, 516, 518, 520, 522, 
         524, 526, 528, 530, 532, 534, 536, 538, 540, 542, 544, 546, 548, 
         550, 552, 554, 556, 558, 560, 562, 564, 566, 568, 570, 572, 574, 
         576, 578, 580, 582, 584, 586, 588, 590, 592, 594, 596, 598, 600, 
         602, 604, 606, 608, 610, 612, 614, 616, 618, 620, 622, 624, 626, 
         628, 630, 632, 634, 636, 638, 640, 642, 644, 646, 648, 650, 652, 
         654, 656, 658, 660, 662, 664, 666, 668, 670, 672, 674, 676, 678, 
         680, 682, 684, 686, 688, 690, 692, 694, 696, 698, 700, 702, 704, 
         706, 708, 710, 712, 714, 716, 718, 720, 722, 724, 726, 728, 730, 
         732, 734, 736, 738, 740, 742, 744, 746, 748, 750, 752, 754, 756, 
         758, 760, 762, 764, 766, 768, 770, 772, 774, 776, 778, 780, 782, 
         784, 786, 788, 790, 792, 794, 796, 798, 800, 802, 804, 806, 808, 
         810, 812, 814, 816, 818, 820, 822, 824, 826, 828, 830, 832, 834, 
         836, 838, 840, 842, 844, 846, 848, 850, 852, 854, 856, 858, 860, 
         862, 864, 866, 868, 870, 872, 874, 876, 878, 880, 882, 884, 886, 
         888, 890, 892, 894, 896, 898, 900, 902, 904, 906, 908, 910, 912, 
         914, 916, 918, 920, 922, 924, 926, 928, 930, 932, 934, 936, 938, 
         940, 942, 944, 946, 948, 950, 952, 954, 956, 958, 960, 962, 964, 
         966, 968, 970, 972, 974, 976, 978, 980, 982, 984, 986, 988, 990, 
         992, 994, 996, 998, 1000, 1002, 1004, 1006, 1008, 1010, 1012, 1014, 
         1016, 1018, 1020, 1022, 1024, 1026, 1028, 1030, 1032, 1034, 1036, 
         1038, 1040, 1042, 1044, 1046, 1048, 1050, 1052, 1054, 1056, 1058, 
         1060, 1062, 1064, 1066, 1068, 1070, 1072, 1074, 1076, 1078, 1080, 
         1082, 1084, 1086, 1088, 1090, 1092, 1094, 1096, 1098, 1100, 1102, 
         1104, 1106, 1108, 1110, 1112, 1114, 1116, 1118, 1120, 1122, 1124, 
         1126, 1128, 1130, 1132, 1134, 1136, 1138, 1140, 1142, 1144, 1146, 
         1148, 1150, 1152, 1154, 1156, 1158, 1160, 1162, 1164, 1166, 1168, 
         1170, 1172, 1174, 1176, 1178, 1180, 1182, 1184, 1186, 1188, 1190, 
         1192, 1194, 1196, 1198, 1200, 1202, 1204, 1206, 1208, 1210, 1212, 
         1214, 1216, 1218, 1220, 1222, 1224, 1226, 1228, 1230, 1232, 1234, 
         1236, 1238, 1240, 1242, 1244, 1246, 1248, 1250, 1252, 1254, 1256, 
         1258, 1260, 1262, 1264, 1266, 1268, 1270, 1272, 1274, 1276, 1278, 
         1280, 1282, 1284, 1286, 1288, 1290, 1292, 1294, 1296, 1298, 1300, 
         1302, 1304, 1306, 1308, 1310, 1312}, {281.8045101424805, 
         -8.399785922320675*^13, 278.32405098662565, -7.89380365572778*^13, 
         275.0484020323772, -7.440159319646156*^13, 271.95684176355627, 
         -7.0313769345395234*^13, 269.0315480568255, -6.661318388419696*^13, 
         265.7086999821292, -6.2600430883843016*^13, 262.5813940380322, 
         -5.900276393474591*^13, 259.6298457535999, -5.5760890023471664*^13, 
         256.83703902410895, -5.282612727038192*^13, 253.66466426025943, 
         -4.964378502124538*^13, 250.67898328087554, -4.679064119156394*^13, 
         247.86110622411977, -4.4219669828369984*^13, 245.19478650157035, 
         -4.189226105670027*^13, 242.1660705285169, -3.936849823276266*^13, 
         239.31560071045604, -3.710581640304774*^13, 236.6253412803189, 
         -3.506691887787966*^13, 234.0797803167636, -3.3221184055190938*^13, 
         231.18821722599736, -3.121972247568755*^13, 228.4668352299828, 
         -2.9425318216744582*^13, 225.89841397688187, 
         -2.7808391051401832*^13, 223.4681429414437, -2.634465499006307*^13, 
         220.70752145964104, -2.475741388331458*^13, 218.10938125602019, 
         -2.3334384301145008*^13, 215.6572804661314, -2.2052106072204668*^13, 
         213.3370781965824, -2.089131805565886*^13, 210.70146834225955, 
         -1.963258010237878*^13, 208.2209886522997, -1.850407312044635*^13, 
         205.87994049416227, -1.7487190842906883*^13, 203.66482228059908, 
         -1.6566657382860299*^13, 201.14856262539922, 
         -1.5568441721522137*^13, 198.78041493164156, 
         -1.4673507531407193*^13, 196.54539013984612, 
         -1.3867098235494492*^13, 194.43059701656856, 
         -1.3137098678005844*^13, 192.0282825229796, -1.234549117984964*^13, 
         189.76737962215418, -1.1635791983494387*^13, 187.6335766931437, 
         -1.0996297737593406*^13, 185.61456521514228, 
         -1.0417400074245494*^13, 183.32103562578746, -9.789642850041592*^12, 
         181.16252070082328, -9.226843330919434*^12, 179.12535557859945, 
         -8.719719699782517*^12, 177.1977880615854, -8.260651724444833*^12, 
         175.00811681315287, -7.762834792129054*^12, 172.94735299904252, 
         -7.316532484368822*^12, 171.00244922704803, -6.914383512815918*^12, 
         169.1621844286539, -6.55034443144532*^12, 167.07166806895682, 
         -6.155574968747154*^12, 165.10422849210565, -5.80165882854598*^12, 
         163.24740781194822, -5.482757896163571*^12, 161.49049203570965, 
         -5.194079266257668*^12, 159.4946401193206, -4.881029642676739*^12, 
         157.61629838736732, -4.60037823236046*^12, 155.8435717627503, 
         -4.3474949468871226*^12, 154.16623036447055, 
         -4.1185786738692715*^12, 152.26075580684676, 
         -3.8703348977322886*^12, 150.46747692996456, -3.647783920694611*^12, 
         148.77503597782257, -3.447253733916873*^12, 147.17366525721968, 
         -3.265729803918943*^12, 145.35447512436707, -3.068878231362807*^12, 
         143.64240684656463, -2.892401691226756*^12, 142.02661565523616, 
         -2.733388019605981*^12, 140.49777511387944, -2.5894465062801685*^12, 
         138.76096183003145, -2.4333495179619126*^12, 137.1264263538378, 
         -2.293410304683948*^12, 135.5838136724852, -2.167319326578392*^12, 
         134.12421862298868, -2.0531806153634463*^12, 132.46605157656296, 
         -1.929402164240146*^12, 130.9055376625642, -1.8184370131219233*^12, 
         129.4327894443891, -1.7184535539412783*^12, 128.03930395438132, 
         -1.627948186510336*^12, 126.45622148341555, -1.52979820863884*^12, 
         124.96637690699359, -1.4418092304923003*^12, 123.56032918962542, 
         -1.3625286824902847*^12, 122.22995934479901, 
         -1.2907641406160664*^12, 120.7185610858153, -1.2129372362165298*^12, 
         119.296185435918, -1.1431679688958752*^12, 117.95381754460304, 
         -1.0803043756625817*^12, 116.68370501719544, 
         -1.0234008162507688*^12, 115.24074460064108, -9.616896960891185*^11, 
         113.8827824047443, -9.063681893671744*^11, 112.60121046340332, 
         -8.565227359095359*^11, 111.38862637199676, -8.114034105142688*^11, 
         110.01100444800193, -7.624716380956725*^11, 108.71453860799227, 
         -7.186067020390829*^11, 107.49100934376116, -6.790841118346853*^11, 
         106.33334839081262, -6.433091557085773*^11, 105.01810597127968, 
         -6.045109163918575*^11, 103.78035149704004, -5.697305330576448*^11, 
         102.6122363257706, -5.3839345944252936*^11, 101.50701120098091, 
         -5.100280762592511*^11, 100.25132330291656, -4.7926524271290826*^11, 
         99.06962132945597, -4.516884168433708*^11, 97.95441070885008, 
         -4.268419669681778*^11, 96.89924674579426, -4.043518920200027*^11, 
         95.700416322704, -3.799606687514753*^11, 94.57222839840027, 
         -3.5809579826733813*^11, 93.50752643702222, -3.3839595451973584*^11, 
         92.50015651189759, -3.205645518069039*^11, 91.35560865995117, 
         -3.0122557819519666*^11, 90.27851129341033, -2.8388985688179254*^11, 
         89.26203060379481, -2.6827085074025168*^11, 88.30029026514838, 
         -2.5413336726646683*^11, 87.20756669148615, -2.38800424920007*^11, 
         86.17924614514202, -2.2505592513881122*^11, 85.20880292979687, 
         -2.1267264160634003*^11, 84.2906257486408, -2.0146406847894748*^11, 
         83.24737949064934, -1.893075431284941*^11, 82.26562681054885, 
         -1.784105269107076*^11, 81.3391361708575, -1.685928291699809*^11, 
         80.46254930168524, -1.597065537430335*^11, 79.46653968509794, 
         -1.5006859842627524*^11, 78.52924595548606, -1.4142931948302612*^11, 
         77.6447174128003, -1.336458269855758*^11, 76.80783735540251, 
         -1.2660084290653362*^11, 75.85692518081405, -1.1895982098962529*^11, 
         74.96207699369646, -1.1211064662480162*^11, 74.11761021332984, 
         -1.0594001027207811*^11, 73.31863876660258, -1.0035491439313402*^11, 
         72.41078171392336, -9.429720201409346*^10, 71.55645684375651, 
         -8.886733315138496*^10, 70.75023755258745, -8.397545603641211*^10, 
         69.98745795154647, -7.95478276232586*^10, 69.12070619283263, 
         -7.474545252740204*^10, 68.30506946732145, -7.04408763514485*^10, 
         67.5353655564781, -6.6562849590403496*^10, 66.80713878439691, 
         -6.305289791323801*^10, 65.97963079499537, -5.9245806803636284*^10, 
         65.20093015256542, -5.583339907554982*^10, 64.4660879563191, 
         -5.275917688117821*^10, 63.77084922362053, -4.99767737793488*^10, 
         62.980807783215724, -4.695877026401832*^10, 62.23737050920873, 
         -4.425368847807735*^10, 61.53581125254372, -4.1816728573499626*^10, 
         60.87206663529396, -3.961112540359545*^10, 60.117795010088415, 
         -3.7218718845114174*^10, 59.408024143438645, -3.507440511665185*^10, 
         58.73824055052705, -3.3142660738499836*^10, 58.104563781185384, 
         -3.139433409723384*^10, 57.38444207893347, -2.9497896885334496*^10, 
         56.70681298153538, -2.779815041415188*^10, 56.06736603776857, 
         -2.626692717018624*^10, 55.462395441242684, -2.48811127684202*^10, 
         54.774877131332964, -2.3377868218621624*^10, 54.12793421275332, 
         -2.2030558958687763*^10, 53.51745007339472, -2.0816850537368652*^10, 
         52.939885641837975, -1.971841579219762*^10, 52.28349423288852, 
         -1.8526877755002235*^10, 51.66584782333383, -1.7458961574993183*^10, 
         51.083014862104854, -1.6496958191454594*^10, 50.53161546212939, 
         -1.5626336422879456*^10, 49.9049413303594, -1.4681900108870474*^10, 
         49.3152646955357, -1.3835466037195122*^10, 48.758830687110105, 
         -1.3072992856139805*^10, 48.23241139369823, -1.2382959329220957*^10, 
         47.634108754544165, -1.163440504519689*^10, 47.0711352453602, 
         -1.0963540950215624*^10, 46.5399046751477, -1.0359233969408836*^10, 
         46.03733422602766, -9.812349506477346*^9, 45.46611824319444, 
         -9.219070322709276*^9, 44.928638574801745, -8.68737760114776*^9, 
         44.42147007079113, -8.208444327774288*^9, 43.94166843630001, 
         -7.775028288634079*^9, 43.39631246155651, -7.304831769873831*^9, 
         42.88317211545584, -6.883454683073992*^9, 42.39897599622135, 
         -6.503898533903365*^9, 41.940912059095616, -6.160421567189998*^9, 
         41.42024499722284, -5.78778672150054*^9, 40.930341741155175, 
         -5.453849831244932*^9, 40.4680776750254, -5.153062248792932*^9, 
         40.03076701537297, -4.88087190448422*^9, 39.53367080832438, 
         -4.585567103349635*^9, 39.065952328370926, -4.32093631350055*^9, 
         38.624627098510814, -4.082580463562004*^9, 38.207129879009, 
         -3.866890746994787*^9, 37.732537104086504, -3.6328778795945516*^9, 
         37.285998744075236, -3.423178080717559*^9, 36.86466411484426, 
         -3.23430375746284*^9, 36.466083061767335, -3.063393972686673*^9, 
         36.01297463846067, -2.8779595679923*^9, 35.58665724165255, 
         -2.71179575225955*^9, 35.184407921503684, -2.562137615254494*^9, 
         34.80388639710707, -2.426717159516168*^9, 34.371289398043054, 
         -2.279783479188757*^9, 33.96427724679131, -2.148123192647975*^9, 
         33.580248943628185, -2.029544371813994*^9, 33.21696910601066, 
         -1.9222491154157758*^9, 32.803954666693976, -1.8058279495360389*^9, 
         32.41537351510516, -1.7015116712199905*^9, 32.048741079434585, 
         -1.6075626353106184*^9, 31.701922125491652, -1.5225556359023004*^9, 
         31.307603448960293, -1.4303152497669337*^9, 30.936618644874166, 
         -1.3476682047988255*^9, 30.586594297248954, -1.2732370222395225*^9, 
         30.255490785375578, -1.2058919651473997*^9, 29.879021237101366, 
         -1.1328137634114833*^9, 29.524835929015772, -1.0673381622227635*^9, 
         29.19066756765757, -1.0083732255010093*^9, 28.874567816326422, 
         -9.55023455684869*^8, 28.515139105484756, -8.971297823366436*^8, 
         28.17699253076552, -8.452608281407961*^8, 27.85796211590304, 
         -7.98551032549997*^8, 27.556186674831068, -7.562905919046662*^8, 
         27.213027117840184, -7.104289319769366*^8, 26.890192968368094, 
         -6.693413878769248*^8, 26.585614979668, -6.323418633740432*^8, 
         26.297515170151136, -5.988677604878647*^8, 25.969888032880952, 
         -5.625396605125966*^8, 25.66167289474033, -5.299945029170859*^8, 
         25.370892858619907, -5.0068846248402023*^8, 25.095849379992238, 
         -4.7417563298471427*^8, 24.78305129511989, -4.454010765268837*^8, 
         24.488793159010843, -4.1962393046002495*^8, 24.211186242698798, 
         -3.964131526417715*^8, 23.948607841949404, -3.754153373002485*^8, 
         23.64996729790315, -3.5262523035808194*^8, 23.369034136937117, 
         -3.3221000210966414*^8, 23.10400380613737, -3.138280150620008*^8, 
         22.853326007715125, -2.971991855854058*^8, 22.56820190606096, 
         -2.791501279315415*^8, 22.29999031797344, -2.629826142897484*^8, 
         22.04696705532759, -2.4842587748696643*^8, 21.807650948476283, 
         -2.3525795225458798*^8, 21.53543122669505, -2.209646770375848*^8, 
         21.27936513759999, -2.0816201170944664*^8, 21.037805219232077, 
         -1.966353657967706*^8, 20.809336300283483, -1.8620883975353417*^8, 
         20.549436616840428, -1.7489066015210494*^8, 20.304966043618165, 
         -1.6475331541880774*^8, 20.07435037102363, -1.5562674326391202*^8, 
         19.856237438063264, -1.4737155550626293*^8, 19.60809991711022, 
         -1.384099156638569*^8, 19.37469978591636, -1.3038365721780212*^8, 
         19.154532770815734, -1.2315801180961439*^8, 18.94630686846292, 
         -1.166225289044426*^8, 18.709398901308102, -1.0952735770350637*^8, 
         18.48656791951608, -1.031730896764247*^8, 18.27637641914513, 
         -9.745293021647893*^7, 18.077589831061868, -9.227936293438742*^7, 
         17.851402931938292, -8.666240599553882*^7, 17.63866251118601, 
         -8.163227062922472*^7, 17.43799481181845, -7.710433328183842*^7, 
         17.248220098857367, -7.300925302292423*^7, 17.032268812551447, 
         -6.856293636116703*^7, 16.829162040595644, -6.458137166412944*^7, 
         16.637586887130073, -6.099750676811531*^7, 16.45641596906864, 
         -5.775640433613173*^7, 16.250236827962844, -5.4237082852341816*^7, 
         16.056327487044467, -5.108581028313067*^7, 15.87343315649246, 
         -4.824945991751174*^7, 15.700476435310843, -4.5684508425507225*^7, 
         15.50362696368176, -4.289919007362354*^7, 15.318498593366025, 
         -4.040531733287626*^7, 15.143892010315513, -3.8160799071433134*^7, 
         14.978777533192213, -3.613115649625138*^7, 14.790835296598528, 
         -3.392698138688617*^7, 14.614090299052906, -3.195357429945309*^7, 
         14.447396191185602, -3.0177589159919202*^7, 14.28976885139401, 
         -2.8571713797737252*^7, 14.110330549080894, -2.6827616886993293*^7, 
         13.941589334841161, -2.5266228264460877*^7, 13.782449426665238, 
         -2.3861131502714578*^7, 13.631970200632331, -2.2590692775642414*^7, 
         13.460650799075832, -2.121079722476453*^7, 13.299550971529918, 
         -1.99755467620069*^7, 13.147623214653434, -1.8864015807643153*^7, 
         13.003968433593048, -1.7859069441591136*^7, 12.840400339236483, 
         -1.6767451625886276*^7, 12.686595915997458, -1.5790335047145886*^7, 
         12.541553754220683, -1.4911144700289488*^7, 12.404414408715574, 
         -1.4116309922297377*^7, 12.248246678179747, -1.325285150709744*^7, 
         12.101407347727436, -1.2480023900340486*^7, 11.962939015421686, 
         -1.1784699021099411*^7, 11.832020091501873, -1.1156129830521334*^7, 
         11.682917677549366, -1.0473231341648499*^7, 11.542728089523163, 
         -9.862062512326421*^6, 11.410535941770574, -9.312226085006071*^6, 
         11.285555787045238, -8.815211780624118*^6, 11.143198818687106, 
         -8.275188877164291*^6, 11.009357906312497, -7.791930482549821*^6, 
         10.883157779373997, -7.357202710860177*^6, 10.7638474978608, 
         -6.964267044036559*^6, 10.627930593042622, -6.537287784337574*^6, 
         10.500150926221167, -6.155225247320267*^6, 10.379671526945657, 
         -5.811559279718183*^6, 10.265774401285652, -5.500954703631905*^6, 
         10.136006010673103, -5.163405491258101*^6, 10.014013178343122, 
         -4.861394039567815*^6, 9.898995501203443, -4.589757658815652*^6, 
         9.790266441017865, -4.344272624892077*^6, 9.666368221520154, 
         -4.0774638225150863*^6, 9.54990024200367, -3.838768883558773*^6, 
         9.440097012544557, -3.624100307944992*^6, 9.336302027781857, 
         -3.4301145871671685*^6, 9.21800824438865, -3.21925525563621*^6, 
         9.106815002388297, -3.030633990310402*^6, 9.001990145828778, 
         -2.8610148455366935*^6, 8.902905843915134, -2.7077513474859986*^6, 
         8.78996279860746, -2.541137207243793*^6, 8.683805507684026, 
         -2.3921110868615177*^6, 8.583733641568326, -2.2581114268854624*^6, 
         8.48914674731315, -2.1370436857430506*^6, 8.381312233840674, 
         -2.005414000802177*^6, 8.279962923233398, -1.887692730130561*^6, 
         8.184428873055218, -1.7818523524728022*^6, 8.094135770293338, 
         -1.686235326375399*^6, 7.991178553566903, -1.5822636073569814*^6, 
         7.894419578183403, -1.4892889384531584*^6, 7.803217914880123, 
         -1.405706663386393*^6, 7.7170242088097645, -1.3302052119065372*^6, 
         7.618723527870518, -1.2480959234534912*^6, 7.526347100463997, 
         -1.174680437015629*^6, 7.439281698849593, -1.1086889037882953*^6, 
         7.3570017981814795, -1.0490836327573096*^6, 7.263146891648988, 
         -984252.8666983438, 7.174954636151138, -926293.9739086209, 
         7.091838253315382, -874202.1696085788, 7.013294971313889, 
         -827156.5516968824, 6.923684624250648, -775979.204982603, 
         6.839487149262437, -730232.7249845666, 6.760141021996571, 
         -689122.1565220974, 6.685165195525303, -651998.1795132483, 
         6.599607306806449, -611607.8035443187, 6.519223798385351, 
         -575508.7224052337, 6.443477258817916, -543072.0699746902, 
         6.371907384614454, -513784.3140697062, 6.290218553832416, 
         -481914.6764435694, 6.213476386664939, -453435.2691645601, 
         6.141166495249543, -427848.67941513326, 6.072848382618782, 
         -404748.8240858874, 5.994853515633638, -379608.5015365589, 
         5.92158788175605, -357146.03965776204, 5.852559076820828, 
         -336968.05500249926, 5.78734551599901, -318753.4791960869, 
         5.712877448292311, -298926.6052196658, 5.642931002566233, 
         -281214.48964540166, 5.577034765675733, -265306.066509665, 
         5.51478521115572, -250947.52376374367, 5.443684348178581, 
         -235315.24430356995, 5.376906869759493, -221352.6841451518, 
         5.314001406169486, -208813.90677141558, 5.254581674306205, 
         -197498.30077111573, 5.186695648024996, -185176.62417844436, 
         5.122943717074857, -174172.99740357543, 5.062893650574249, 
         -164293.00789598798, 5.006175630803143, -155378.12102576735, 
         4.941358971719018, -145668.74598649284, 4.880495660680602, 
         -136999.5817437572, 4.8231717421753055, -129216.97113794237, 
         4.76903312123246, -122195.65370948875, 4.707146945169077, 
         -114547.06976701992, 4.649041523937458, -107719.22918800169, 
         4.5943203531486505, -101590.70789507806, 4.542644351698443, 
         -96062.55464614436, 4.483556060651548, -90039.27310846199, 
         4.428083714979451, -84663.40990595824, 4.375847474631457, 
         -79839.03018358113, 4.3265225957116344, -75487.98164647805, 
         4.270105592134637, -70746.20074130342, 4.217147154681095, 
         -66514.98253041413, 4.167283356620245, -62718.545456568194, 
         4.1202031453747345, -59295.17731125218, 4.066336559291468, 
         -55563.53874454317, 4.01577825273989, -52234.426490065496, 
         3.9681794954419205, -49247.9908731023, 3.9232423096274265, 
         -46555.512180329715, 3.8718107379350464, -43619.88605034465, 
         3.8235439295835474, -41001.51967354414, 3.7781076664880002, 
         -38653.15599145271, 3.7352164572247974, -36536.3413316068, 
         3.6861097146633717, -34227.79793709528, 3.640030681991873, 
         -32169.234591598375, 3.596659000193443, -30323.345747470983, 
         3.555721102513539, -28659.787929047347, 3.5088339837373814, 
         -26845.090460711577, 3.464843690420835, -25227.29913331963, 
         3.423443099217719, -23776.971291570724, 3.3843700319380785, 
         -22470.169263069332, 3.3396020841660228, -21044.260295705208, 
         3.297605966041663, -19773.40081484827, 3.258087194876448, 
         -18634.359597654686, 3.220794469357767, -17608.25751188223, 
         3.1780497751209107, -16488.32220598118, 3.1379575356551728, 
         -15490.435245794994, 3.1002353410183727, -14596.271244006353, 
         3.0646422784017506, -13790.944824848879, 3.0238292478898456, 
         -12911.720819930506, 2.9855546626765546, -12128.533464352717, 
         2.949547643531673, -11426.931208296373, 2.9155772000365694, 
         -10795.181050899264, 2.8766083726029446, -10105.253961657727, 
         2.840069102477329, -9490.867116415722, 2.805699523814294, 
         -8940.627209743605, 2.77327812372489, -8445.28888876242, 
         2.7360699781025333, -7904.166870560955, 2.701187390450359, 
         -7422.439660703281, 2.668381014570586, -6991.1277481340185, 
         2.637438390535423, -6602.948369077555, 2.6019111633413132, 
         -6178.752397275289, 2.568610161205144, -5801.237347456721, 
         2.5372960863626415, -5463.328381011147, 2.507765126652195, 
         -5159.2906721889985, 2.4738426387931955, -4826.931348335377, 
         2.4420514974107035, -4531.244733932521, 2.41216200346373, 
         -4266.6585324443495, 2.3839786058582497, -4028.6584679827265, 
         2.351588096426421, -3768.3972013127304, 2.3212383068144113, 
         -3536.9331613921418, 2.2927087075244073, -3329.8790381043254, 
         2.265811639489466, -3143.682674998736, 2.234883606800362, 
         -2939.996478341356, 2.205909726057014, -2758.9126295568717, 
         2.1786782277245558, -2596.9780925133746, 2.153008992579796, 
         -2451.398882121108, 2.1234770419780573, -2292.0849417235418, 
         2.0958165499512713, -2150.502377571911, 2.069824116058027, 
         -2023.9345347950552, 2.0453268248270673, -1910.1846887668955, 
         2.0171275229384635, -1785.654251894933, 1.9907206849574952, 
         -1675.026790953575, 1.9659109065277793, -1576.1658699959728, 
         1.9425321551978445, -1487.3451032949902, 1.9156048902868443, 
         -1390.0668120570672, 1.8903946256379764, -1303.6838314251952, 
         1.8667135970200655, -1226.5167446927137, 1.8444023489290027, 
         -1157.2094213078926, 1.8186891970606005, -1081.2705913670964, 
         1.7946209529236938, -1013.8652716713867, 1.7720171527241622, 
         -953.6738966560881, 1.7507246258230242, -899.6316934538728, 
         1.726170222531726, -840.3926538440384, 1.703191853099578, 
         -787.8333833853138, 1.681616030009731, -740.9175529233588, 
         1.6612955887555274, -698.809570539313, 1.6378470059501833, 
         -652.6314121738897, 1.6159086564768628, -611.6787782104813, 
         1.595313719753265, -575.1381854012807, 1.5759207714082681, 
         -542.3542391915281, 1.553527399244292, -506.3844543849993, 
         1.5325813947730356, -474.4999484405239, 1.5129223091398551, 
         -446.0624972976564, 1.4944142042606587, -420.5583271897333, 
         1.4730276377510734, -392.56209007708486, 1.4530283763096024, 
         -367.7575769632592, 1.4342620610855392, -345.6443380898822, 
         1.4165979980182213, -325.81985085358224, 1.396171928167674, 
         -304.04727024793607, 1.377075778232709, -284.7665794973986, 
         1.3591610105002092, -267.5855861541178, 1.3423019437137391, 
         -252.1891192371981, 1.3227920529880561, -235.27083491469457, 
         1.3045572550492115, -220.29665715550303, 1.2874545767120544, 
         -206.9594167555408, 1.2713631288304363, -195.01249402996257, 
         1.2527269908194671, -181.87759820559648, 1.235313562916647, 
         -170.25830930749257, 1.2189851915348853, -159.91420008181848, 
         1.2036255689705486, -150.6524193027011, 1.1858225521370631, 
         -140.46395697619775, 1.169192199281701, -131.45612995184132, 
         1.1536019426101742, -123.44087461044131, 1.1389398547392158, 
         -116.2674894952495, 1.1219310301998733, -108.37179550348402, 
         1.1060470576441142, -101.39505778544896, 1.0911602318549258, 
         -95.19027270179133, 1.077162813729325, -89.63976103192998, 
         1.0609108671022185, -83.5266842253315, 1.0457380975066517, 
         -78.12828633964085, 1.031521449160861, -73.32974133400374, 
         1.0181571878362543, -69.03922524816966, 1.0026263352568925, 
         -64.31091561369067, 0.988131029878129, -60.137936167277445, 
         0.9745526617853603, -56.43066210530076, 0.9617913264187296, 
         -53.11750010204946, 0.9469472349646229, -49.463928067100156, 
         0.9330970191199984, -46.241480832585125, 0.92012632035869, 
         -43.38025829101914, 0.9079388963626435, -40.82448596134336, 
         0.8937486092636844, -38.004252247622034, 0.8805124024695016, 
         -35.51840876541865, 0.868119982981342, -33.31248325682203, 
         0.8564786106638416, -31.343061589421577, 0.8429104778783907, 
         -29.168362820369005, 0.8302584292599811, -27.252778001671373, 
         0.8184160596330984, -25.553893367069634, 0.8072939779516612, 
         -24.037945576334756, 0.7943175929392821, -22.362802700466286, 
         0.7822210227618704, -20.888246901679306, 0.7709015800675166, 
         -19.581282410301164, 0.7602730763764053, -18.41567522137213, 
         0.7478592202150418, -17.126722400441363, 0.7362905687007001, 
         -15.992893605910892, 0.7254679895635909, -14.988543062511688, 
         0.715308356546758, -14.093306202846033, 0.703428950957849, 
         -13.102587951444736, 0.6923617359660065, -12.231711724640686, 
         0.6820109784569767, -11.460763177335043, 0.6722964798442141, 
         -10.773949328362733, 0.6609245511995296, -10.013291921467259, 
         0.6503333368541746, -9.345122302351633, 0.6404303532943617, 
         -8.753991632537154, 0.6311382004531663, -8.227664489224763, 
         0.6202478574855703, -7.644280727507894, 0.6101082364699134, 
         -7.132198300620186, 0.6006299598622721, -6.6794444849948, 
         0.5917383019732583, -6.276549753171984, 0.5813047306924363, 
         -5.8296094155672025, 0.5715933236937405, -5.43757810764975, 
         0.562517671242154, -5.091186546689983, 0.5540056024924248, 
         -4.783113539394097, 0.5440050827433022, -4.441069464107915, 
         0.5346995594365247, -4.141265010265653, 0.5260054574876717, 
         -3.8765313945398923, 0.5178530455602158, -3.641214417971363, 
         0.5082629947239257, -3.3797194515798044, 0.49934212169483083, 
         -3.150682857104149, 0.4910095573996317, -2.9485662536276425, 
         0.4831978984480916, -2.7690076030893573, 0.473996948924871, 
         -2.569293259288495, 0.46544067099030706, -2.3944943181192766, 
         0.45745077695678255, -2.240336603357841, 0.44996208314856334, 
         -2.103458614986595, 0.44113020135004266, -1.9510742524325324, 
         0.4329197636887493, -1.8177951050834187, 0.42525494272343334, 
         -1.7003262037262283, 0.41807266913129487, -1.5960799221312418, 
         0.4095913244866253, -1.4799132103763948, 0.40170944356580685, 
         -1.3783814859809558, 0.39435354097274017, -1.2889473990044777, 
         0.3874625587721391, -1.2096212005726297, 0.379314951358804, 
         -1.1211378767824254, 0.37174604242952214, -1.0438543106371259, 
         0.36468457281834865, -0.9758186602056774, 0.3580713949377072, 
         -0.9155025439385145, 0.3502427495037815, -0.8481569821000505, 
         0.3429732170711965, -0.7893744341458679, 0.33619365077369856, 
         -0.7376550263568719, 0.3298467138412562, -0.6918259789920321, 
         0.32232464853435083, -0.640604707858936, 0.3153432455324449, 
         -0.5959249498283135, 0.30883535798429496, -0.5566353044696235, 
         0.30274536173731836, -0.5218367449834853, 0.2955203660713558, 
         -0.48290547666509, 0.28881864812001273, -0.4489675915231106, 
         0.2825749516087344, -0.4191400620389647, 0.27673526912899266, 
         -0.392734361780233, 0.26980156617407886, -0.3631663430465087, 
         0.26337467047875224, -0.33740691658138217, 0.2573911204032884, 
         -0.31477960642117025, 0.2517984401426413, -0.29475763583324344, 
         0.24515747189864043, -0.27233099247788417, 0.2390071245511296, 
         -0.2528043421420135, 0.2332857302233474, -0.23566070521298882, 
         0.22794233365511912, -0.22049795464901023, 0.22162278524743823, 
         -0.2035735261162999, 0.21577438162272386, -0.18883844325767865, 
         0.2103378774260215, -0.17590251768753162, 0.2052643131241301, 
         -0.16446224226276215, 0.19959043892395564, -0.1523588375631958, 
         0.19432288881873916, -0.14174572951063086, 0.18941296598940194, 
         -0.13236980266329582, 0.1836065384405258, -0.12189632381016313, 
         0.17824656852288784, -0.11279226545754442, 0.17327630902652302, 
         -0.10481186768096809, 0.16864893175649293, -0.09776432821960056, 
         0.16347902299584194, -0.09030808578916247, 0.1586930041344169, 
         -0.08378185111484748, 0.1542442616870348, -0.07802653940983256, 
         0.14899012263793535, -0.0715991048366601, 0.14415682807337685, 
         -0.06602531932240438, 0.13969002724932553, -0.06115072461718554, 
         0.13554494766502237, -0.05685562112112452, 0.13092678221221166, 
         -0.05231936093458449, 0.1266675591627447, -0.04835961800434639, 
         0.12272285070879926, -0.044876874348913, 0.11810154505643637, 
         -0.041014282287209096, 0.11386726976301476, -0.037673678645114804, 
         0.10996913475831573, -0.03476000480178351, 0.10636527527546873, 
         -0.03219963967003013, 0.10302086817536438, -0.02993471338984287, 
         0.09990666356782033, -0.027919146162831608, 0.09699785680560337, 
         -0.026115798973332667, 0.09476340588705373, -0.02478133831980065, 
         0.09264136815446791, -0.023554066769236413, 0.09062285708300079, 
         -0.02242224452829384, 0.08869994679368297, -0.02137575227520344, 
         0.08686552999031258, -0.020405817884876994, 0.08634712742664824, 
         -0.020136682916002648, 0.08583553490464974, -0.019873212471593904, 
         0.08533060969397181, -0.019615240797250126, 0.08483221380448136, 
         -0.019362608737105397, 0.08478272802677013, -0.019337633202459983, 
         0.08473330605509341, -0.01931270937031436, 0.08468394775742595, 
         -0.01928783708572577, 0.08465928451061615, -0.019275416236403747, 
         0.08463463714627523, -0.01926300823428449, 0.08460999025790752, 
         -0.01925060531249827, 0.08458535923281434, -0.01923821520336846, 
         0.0845361288702353, -0.01921346540073684, 0.08448696181745524, 
         -0.019188766691474114, 0.08443785794383821, -0.019164118922875336, 
         0.08395026635514566, -0.018920411274105344, 0.08346885122563254, 
         -0.018681639149674266, 0.08299348846738416, -0.0184476639568014, 
         0.0825240569712956, -0.018218351940580902, 0.08085852426895614, 
         -0.01741864105798013, 0.07926515700280325, -0.016673651658010175, 
         0.07773908980644663, -0.01597829695242384, 0.07627589350297703, 
         -0.015328080252602638, 0.07487153771332528, -0.014719018645894397, 
         0.07287121152526967, -0.013876545794148305, 0.07098351810785117, 
         -0.013108144888522398, 0.06919870727012033, -0.012405062156684599, 
         0.06750817611463117, -0.011759838472957366, 0.06590428532802127, 
         -0.0111660780487148, 0.0643802270130384, -0.010618265773735735, 
         0.062355805161417664, -0.009914986746712763, 0.060463013906652284, 
         -0.009282264466858552, 0.05868888918525584, -0.00871068890804661, 
         0.05702216437069062, -0.008192402491473414, 0.05545299563920343, 
         -0.007720796798400629, 0.05397273953698117, -0.00729027809635823, 
         0.052202552550363594, -0.006793535461538671, 0.050550769413770506, 
         -0.006347592250651, 0.04900550252651518, -0.005945585716971972, 
         0.047556440840947796, -0.00558179493703475, 0.04619459295667531, 
         -0.005251416562860761, 0.04491207887619565, -0.004950391635515937, 
         0.04336850174015632, -0.004600967908611287, 0.0419318526392677, 
         -0.0042882656900425425, 0.04059110070410318, -0.004007215996943949, 
         0.03933670980219885, -0.0037536080346494186, 0.038160390692203304, 
         -0.003523918180328466, 0.03705490082149813, -0.003315178441213198, 
         0.0357239774273981, -0.003072952391558497, 0.034488461825119464, 
         -0.0028568986681232742, 0.03333823648809853, -0.002663321091780177, 
         0.03226458633008481, -0.0024891613405211566, 0.031259961710437094, 
         -0.0023318697217764137, 0.030317787742144147, 
         -0.0021893062441246097, 0.029181627451109592, 
         -0.0020237217226735846, 0.028129743954080275, 
         -0.0018765506797533166, 0.02715294166697134, -0.0017451281649583877, 
         0.02624333100096108, -0.0016272590035390287, 0.0253941029805816, 
         -0.0015211208004840748, 0.02459934872258403, -0.001425190107943557, 
         0.023638726035694887, -0.0013136021911604918, 0.022751836786332705, 
         -0.0012147969582769456, 0.02193039620585573, -0.0011268757059907158, 
         0.02116732678535755, -0.0010482822609909693, 0.020456545132073702, 
         -0.000977730747587195, 0.019792792137286765, -0.000914150967689612, 
         0.018988054930248313, -0.0008400376679860368, 0.01824724501574692, 
         -0.0007746778703796778, 0.01756295935049078, -0.0007167353230348275, 
         0.017130370975971403, -0.0006813051870530186, 0.016718901045520547, 
         -0.0006484645898001222}}, {Automatic}]}}
 
A1[nq_, Q2_, fi_] := (1/Pi)*NIntegrate[tr12d[nq, Exp[w]]/
        (1 + (Q2/LL2[nq])*Exp[I*fi]*Exp[-w]), {w, lns0[nq], lnsm}, 
       MaxRecursion -> 30, WorkingPrecision -> 16] + 
     f1ss0[nq]/((Q2/LL2[nq])*Exp[I*fi] + s1s0[nq]) + 
     f2ss0[nq]/((Q2/LL2[nq])*Exp[I*fi] + s2s0[nq]) + J0[nq] - 
     (Q2/LL2[nq])*Exp[I*fi]*J1[nq] + (Q2/LL2[nq])^2*Exp[I*2*fi]*J2[nq]
 
A1 /: A1::usage = "A1[Nf,Q2,fi] = A(Q^2) = A_1(Q^2), where Q^2=Q2 Exp[I fi] \
(Q2=|Q^2|, in GeV^2) and A(Q^2) is the 2danQCD analog of the underlying \
perturbative coupling a(Q^2) = a2d[Nf,Q2/LL2[Nf],fi] (see: a2d); at high \
|Q^2|> 1 GeV^2 the two couplings practically coincide"
 
tr12d[nq_, s_] := If[s != 0, If[s > 0, Im[a2d[nq, s, -Pi]], 
      Im[(1/c1[nq])*(1/(1 - c22d[nq]/c1[nq]^2 + ProductLog[-1, 
           zu2d[nq, -s, 0]]))]], 0]
 
tr12d /: tr12d::usage = "tr12d[Nf,s] is the discontinuity function \
rho_1^{(pt)}(sigma)=Im a(Q^2=-sigma - I epsilon) of the underlying pQCD \
coupling a(Q^2) (see: a2d) in the considered Nf-flavor 2danQCD, where \
s=sigma/LL2[Nf] (see: LL2)"
 
lns0[3] = 3.242592351485517
 
lns0[4] = 3.242592351485517
 
lns0[5] = 3.242592351485517
 
lns0[6] = 3.242592351485517
 
lnsm = 92.10340371976183
 
J0[3] = 0.00466999455549764309108360437157522531`16.
 
J0[4] = 0.00507753887074683231808795530064138381`16.
 
J0[5] = 0.00557750058581599984275926991902285832`16.
 
J0[6] = 0.00622202997429364348515952541839318206`16.
 
J1[3] = 4.84346028881497324758771242290045`16.*^-45
 
J1[4] = 5.29827956855580615903950025463651`16.*^-45
 
J1[5] = 5.87569345738507775148605975808154`16.*^-45
 
J1[6] = 6.66537630861582027402462127582015`16.*^-45
 
J2[3] = 2.446809702456800300219863101055`16.*^-85
 
J2[4] = 2.676734109529465657890833417244`16.*^-85
 
J2[5] = 2.968727412713396780222296111734`16.*^-85
 
J2[6] = 3.368285035422341381188279148526`16.*^-85
 
tA2[nq_, Q2_, fi_] := (1/be0[nq]/Pi)*((Q2/LL2[nq])*Exp[I*fi])*
     (NIntegrate[Exp[-w]*(tr12d[nq, Exp[w]]/
         (1 + (Q2/LL2[nq])*Exp[I*fi]*Exp[-w])^2), {w, lns0[nq], lnsm}, 
       MaxRecursion -> 30, WorkingPrecision -> 16] + 
      (f1ss0[nq]*Pi)/((Q2/LL2[nq])*Exp[I*fi] + s1s0[nq])^2 + 
      (f2ss0[nq]*Pi)/((Q2/LL2[nq])*Exp[I*fi] + s2s0[nq])^2 + J1[nq] - 
      2*(Q2/LL2[nq])*Exp[I*fi]*J2[nq] + 3*(Q2/LL2[nq])^2*Exp[I*2*fi]*J3[nq])
 
tA2 /: tA2::usage = "tA2[Nf,Q2,fi]={tilde A}_2(Q^2) for Q^2=Q2 Exp[I fi] \
(Q2=|Q^2|, in GeV^2); this is the first logarithmic derivative of A_1 (see: \
A1): {tilde A}_2(Q^2) = (-1)/(beta_0 1!) (d/d ln Q^2) A_1(Q^2)"
 
J3[3] = 1.63689500729422241956655214488`16.*^-125
 
J3[4] = 1.79074905870213668885243599517`16.*^-125
 
J3[5] = 1.98615801578900018756979502896`16.*^-125
 
J3[6] = 2.25360240588900587649899107709`16.*^-125
 
tA3[nq_, Q2_, fi_] := (1/2/be0[nq]^2/Pi)*((Q2/LL2[nq])*Exp[I*fi])*
     (NIntegrate[Exp[-w]*(tr12d[nq, Exp[w]]/
         (1 + (Q2/LL2[nq])*Exp[I*fi]*Exp[-w])^2), {w, lns0[nq], lnsm}, 
       MaxRecursion -> 30, WorkingPrecision -> 16] - 
      2*NIntegrate[Exp[-w]*(tr12d[nq, Exp[w]]/
          (1 + (Q2/LL2[nq])*Exp[I*fi]*Exp[-w])^3), {w, lns0[nq], lnsm}, 
        MaxRecursion -> 30, WorkingPrecision -> 16] + 
      (f1ss0[nq]*Pi)*((-s1s0[nq] + (Q2/LL2[nq])*Exp[I*fi])/
        (s1s0[nq] + (Q2/LL2[nq])*Exp[I*fi])^3) + (f2ss0[nq]*Pi)*
       ((-s2s0[nq] + (Q2/LL2[nq])*Exp[I*fi])/
        (s2s0[nq] + (Q2/LL2[nq])*Exp[I*fi])^3))
 
tA3 /: tA3::usage = "tA3[Nf,Q2,fi]={tilde A}_3(Q^2) for Q^2=Q2 Exp[I fi]; \
this is the second logarithmic derivative of A_1 (see: A1): {tilde A}_3(Q^2) \
= (-1)^2/(beta_0^2 2!) (d/d ln Q^2)^2 A_1(Q^2)"
 
tA4[nq_, Q2_, fi_] := (1/6/be0[nq]^3/Pi)*((Q2/LL2[nq])*Exp[I*fi])*
     (NIntegrate[Exp[-w]*(1 - 4*Exp[-w]*(Q2/LL2[nq])*Exp[I*fi] + 
         Exp[-2*w]*(Q2/LL2[nq])^2*Exp[2*I*fi])*(tr12d[nq, Exp[w]]/
         (1 + (Q2/LL2[nq])*Exp[I*fi]*Exp[-w])^4), {w, lns0[nq], lnsm}, 
       MaxRecursion -> 30, WorkingPrecision -> 16] + 
      (f1ss0[nq]*Pi)*((s1s0[nq]^2 - 4*s1s0[nq]*(Q2/LL2[nq])*Exp[I*fi] + 
         (Q2/LL2[nq])^2*Exp[2*I*fi])/(s1s0[nq] + (Q2/LL2[nq])*Exp[I*fi])^4) + 
      (f2ss0[nq]*Pi)*((s2s0[nq]^2 - 4*s2s0[nq]*(Q2/LL2[nq])*Exp[I*fi] + 
         (Q2/LL2[nq])^2*Exp[2*I*fi])/(s2s0[nq] + (Q2/LL2[nq])*Exp[I*fi])^4))
 
tA4 /: tA4::usage = "tA4[Nf,Q2,fi]={tilde A}_4(Q^2) for Q^2=Q2 Exp[I fi]; \
this is the third logarithmic derivative of A_1 (see: A1): {tilde A}_4(Q^2) = \
(-1)^3/(beta_0^3 3!) (d/d ln Q^2)^3 A_1(Q^2)"
 
tA[nq_, nupp_, Q2_, fi_] := tA2d[nq, nupp, Q2, fi]
 
tA2d[nq_, nupp_, Q2_, fi_] := If[Abs[nupp] <= 2., 
     tAMA1l[nq, nupp, Q2/LL2[nq], fi] + 
      (-(Pi*be0[nq]^(nupp - 1)*Gamma[nupp])^(-1))*
       ((Pi*(f1ss0[nq]/s1s0[nq]))*PolyLog[-nupp + 1, (-s1s0[nq])*Exp[(-I)*fi]*
           (LL2[nq]/Q2)] + (Pi*(f2ss0[nq]/s2s0[nq]))*PolyLog[-nupp + 1, 
          (-s2s0[nq])*Exp[(-I)*fi]*(LL2[nq]/Q2)] - 
        NIntegrate[(tr1MA1l[nq, ssig]/ssig)*PolyLog[-nupp + 1, 
           (-ssig)*Exp[(-I)*fi]*(LL2[nq]/Q2)], {ssig, 0, s0[nq]}] + 
        NIntegrate[((tr12d[nq, ssig] - tr1MA1l[nq, ssig])/ssig)*
          PolyLog[-nupp + 1, (-ssig)*Exp[(-I)*fi]*(LL2[nq]/Q2)], 
         {ssig, s0[nq], 10*s0[nq]}] + NIntegrate[
         (tr12d[nq, Exp[w]] - tr1MA1l[nq, Exp[w]])*PolyLog[-nupp + 1, 
           (-Exp[w])*Exp[(-I)*fi]*(LL2[nq]/Q2)], {w, Log[10*s0[nq]], 10.^9}, 
         MaxRecursion -> 30, WorkingPrecision -> 16]), 
     tAMA1l[nq, nupp, Q2/LL2[nq], fi] + 
      (-(Pi*be0[nq]^(nupp - 1)*Gamma[nupp])^(-1))*
       ((Pi*(f1ss0[nq]/s1s0[nq]))*PolyLog[-nupp + 1, (-s1s0[nq])*Exp[(-I)*fi]*
           (LL2[nq]/Q2)] + (Pi*(f2ss0[nq]/s2s0[nq]))*PolyLog[-nupp + 1, 
          (-s2s0[nq])*Exp[(-I)*fi]*(LL2[nq]/Q2)] - 
        NIntegrate[(tr1MA1l[nq, ssig]/ssig)*PolyLog[-nupp + 1, 
           (-ssig)*Exp[(-I)*fi]*(LL2[nq]/Q2)], {ssig, 0, s0[nq]}] + 
        NIntegrate[((tr12d[nq, ssig] - tr1MA1l[nq, ssig])/ssig)*
          PolyLog[-nupp + 1, (-ssig)*Exp[(-I)*fi]*(LL2[nq]/Q2)], 
         {ssig, s0[nq], 10*s0[nq]}] + NIntegrate[
         (tr12d[nq, Exp[w]] - tr1MA1l[nq, Exp[w]])*PolyLog[-nupp + 1, 
           (-Exp[w])*Exp[(-I)*fi]*(LL2[nq]/Q2)], {w, Log[10*s0[nq]], 10.^6}, 
         MaxRecursion -> 30, WorkingPrecision -> 16])]
 
tA2d /: tA2d::usage = "tA2d[Nf,nu,Q2,fi]= {tilde A}_{nu}(Q^2), with Q^2=Q2 \
Exp[I fi]  (Q2=|Q^2|); {tilde A}_{nu} can have nu noninteger; {tilde A}_{nu} \
is the generalization (to noninteger nu) of the logarithmic derivative \
(-1)^{nu-1}/(beta_0^{nu-1} (nu-1)! (d/d ln Q^2)^(nu-1) A_1(Q^2)"
 
tAMA1l[nq_, nu_, u_, fi_] := If[Abs[u*Exp[I*fi] - 1] > Delt[nu], 
     (1/be0[nq]^nu)*(1/(Log[u] + I*fi)^nu - 
       PolyLog[-nu + 1, (1/u)*Exp[(-I)*fi]]/Gamma[nu]), 
     (1/be0[nq]^nu)*Limt1l[u, nu, fi]]
 
Delt[nu_] := 10.^(-2) + (HeavisideTheta[nu - 3]*(nu - 3)*2)/10.^2
 
Limt1l[u_, nu_, fi_] := -(Zeta[1 - nu]/Gamma[nu]) + 
     ((u*Exp[I*fi] - 1)*Zeta[-nu])/Gamma[nu] - 
     ((u*Exp[I*fi] - 1)^2*(Zeta[-1 - nu] + Zeta[-nu]))/(2*Gamma[nu])
 
tr1MA1l[nq_, s_] := (Pi/be0[nq])*(1/(Log[s]^2 + Pi^2))
 
AN4[nq_, M_, nupp_, Q2_, fi_] := A2dN4[nq, M, nupp, Q2, fi]
 
A2dN4[nq_, 4, nupp_, Q2_, fi_] := tA2d[nq, 4 + nupp, Q2, fi]
 
A2dN4[nq_, 3, nupp_, Q2_, fi_] := tA2d[nq, 3 + nupp, Q2, fi] + 
     tk1f[nq, 3 + nupp]*tA2d[nq, 4 + nupp, Q2, fi]
 
A2dN4[nq_, 2, nupp_, Q2_, fi_] := tA2d[nq, 2 + nupp, Q2, fi] + 
     tk1f[nq, 2 + nupp]*tA2d[nq, 3 + nupp, Q2, fi] + 
     tk2f[nq, 2 + nupp]*tA2d[nq, 4 + nupp, Q2, fi]
 
A2dN4[nq_, 1, nupp_, Q2_, fi_] := tA2d[nq, 1 + nupp, Q2, fi] + 
     tk1f[nq, 1 + nupp]*tA2d[nq, 2 + nupp, Q2, fi] + 
     tk2f[nq, 1 + nupp]*tA2d[nq, 3 + nupp, Q2, fi] + 
     tk3f[nq, 1 + nupp]*tA2d[nq, 4 + nupp, Q2, fi]
 
A2dN4[nq_, 0, nupp_, Q2_, fi_] := tA2d[nq, nupp, Q2, fi] + 
     tk1f[nq, nupp]*tA2d[nq, 1 + nupp, Q2, fi] + 
     tk2f[nq, nupp]*tA2d[nq, 2 + nupp, Q2, fi] + 
     tk3f[nq, nupp]*tA2d[nq, 3 + nupp, Q2, fi] + 
     tk4f[nq, nupp]*tA2d[nq, 4 + nupp, Q2, fi]
 
tk1f[nq_, s_] := -k1f[nq, s]
 
k1f[nq_, s_] := c1[nq]*s*(-1 + HarmonicNumber[s])
 
tk2f[nq_, s_] := k1f[nq, s]*k1f[nq, s + 1] - k2f[nq, s]
 
k2f[nq_, s_] := ((s - 1)*(s/2))*c22d[nq] + 
     c1[nq]^2*((s*((s + 1)/2))*((PolyGamma[s + 1] - PolyGamma[1])^2 + 
         (PolyGamma[1, s + 1] - PolyGamma[1, 1]) + 1) - 
       s^2*(PolyGamma[s + 1] - PolyGamma[1]))
 
tk3f[nq_, s_] := (-k1f[nq, s])*k1f[nq, s + 1]*k1f[nq, s + 2] + 
     k1f[nq, s]*k2f[nq, s + 1] + k1f[nq, s + 2]*k2f[nq, s] - k3f[nq, s]
 
k3f[nq_, s_] := (s*(s - 1)*((s + 4)/12))*c32d[nq] + 
     (s*(s + 1)*((s + 2)/2))*c1[nq]*c22d[nq]*((s/(s + 2))*ES[1, s + 1] + 
       2/(s + 1) + 1/(s + 2) - 11/6) + (s*(s + 1)*((s + 2)/6))*c1[nq]^3*
      (Z3[s + 2] - 3*Z2[s + 1] + 3*ES[1, s] - 1)
 
ES[m_, n_] := ((-1)^(m - 1)/(m - 1)!)*(PolyGamma[m - 1, n + 1] - 
      PolyGamma[m - 1, 1])
 
Z3[n_] := ES[1, n]^3 - 3*ES[1, n]*ES[2, n] + 2*ES[3, n]
 
Z2[n_] := ES[1, n]^2 - ES[2, n]
 
tk4f[nq_, s_] := k1f[nq, s]*k1f[nq, s + 1]*k1f[nq, s + 2]*k1f[nq, s + 3] - 
     k1f[nq, s]*k1f[nq, s + 1]*k2f[nq, s + 2] - k1f[nq, s]*k1f[nq, s + 3]*
      k2f[nq, s + 1] - k1f[nq, s + 2]*k1f[nq, s + 3]*k2f[nq, s] + 
     k2f[nq, s]*k2f[nq, s + 2] + k1f[nq, s]*k3f[nq, s + 1] + 
     k1f[nq, s + 3]*k3f[nq, s] - k4f[nq, s]
 
k4f[nq_, s_] := ((s*(s + 1)*(s + 2)*(s + 3))/6)*
     (c42d[nq]*(1/12 - 2/((s + 1)*(s + 2)*(s + 3))) + 
      c22d[nq]^2*(5/12 - (3*s^2 + 6*s + 1)/((s + 1)*(s + 2)*(s + 3))) + 
      (c1[nq]*(c32d[nq]/2))*((1 - 6/((s + 2)*(s + 3)))*Z1[s + 1] + 
        4/(s + 1) - 1/(s + 2) - 1/(s + 3) - 13/6) + (c1[nq]^2*(c22d[nq]/2))*
       ((3*((s + 1)/(s + 3)))*Z2[s + 2] + (12/(s + 2) + 6/(s + 3) - 11)*
         Z1[s + 1] - 6/(s + 1) - 5/(s + 2) - 11/(s + 3) + 38/3) + 
      (c1[nq]^4/4)*(Z4[s + 3] - 4*Z3[s + 2] + 6*Z2[s + 1] - 4*Z1[s] + 1))
 
Z1[n_] := ES[1, n]
 
Z4[n_] := ES[1, n]^4 - 6*ES[1, n]^2*ES[2, n] + 3*ES[2, n]^2 + 
     8*ES[1, n]*ES[3, n] - 6*ES[4, n]
 
AN3[nq_, M_, nupp_, Q2_, fi_] := A2dN3[nq, M, nupp, Q2, fi]
 
A2dN3[nq_, 3, nupp_, Q2_, fi_] := tA2d[nq, 3 + nupp, Q2, fi]
 
A2dN3[nq_, 2, nupp_, Q2_, fi_] := tA2d[nq, 2 + nupp, Q2, fi] + 
     tk1f[nq, 2 + nupp]*tA2d[nq, 3 + nupp, Q2, fi]
 
A2dN3[nq_, 1, nupp_, Q2_, fi_] := tA2d[nq, 1 + nupp, Q2, fi] + 
     tk1f[nq, 1 + nupp]*tA2d[nq, 2 + nupp, Q2, fi] + 
     tk2f[nq, 1 + nupp]*tA2d[nq, 3 + nupp, Q2, fi]
 
A2dN3[nq_, 0, nupp_, Q2_, fi_] := tA2d[nq, nupp, Q2, fi] + 
     tk1f[nq, nupp]*tA2d[nq, 1 + nupp, Q2, fi] + 
     tk2f[nq, nupp]*tA2d[nq, 2 + nupp, Q2, fi] + 
     tk3f[nq, nupp]*tA2d[nq, 3 + nupp, Q2, fi]
 
A5N4[nq_, Q2_, fi_] := tA2d[nq, 5, Q2, fi]
 
A4N4[nq_, Q2_, fi_] := tA4[nq, Q2, fi] + tk1f[nq, 4]*tA2d[nq, 5, Q2, fi]
 
A3N4[nq_, Q2_, fi_] := tA3[nq, Q2, fi] + tk1f[nq, 3]*tA4[nq, Q2, fi] + 
     tk2f[nq, 3]*tA2d[nq, 5, Q2, fi]
 
A2N4[nq_, Q2_, fi_] := tA2[nq, Q2, fi] + tk1f[nq, 2]*tA3[nq, Q2, fi] + 
     tk2f[nq, 2]*tA4[nq, Q2, fi] + tk3f[nq, 2]*tA2d[nq, 5, Q2, fi]
 
A1N4[nq_, Q2_, fi_] := A1[nq, Q2, fi]
 
A4N3[nq_, Q2_, fi_] := tA4[nq, Q2, fi]
 
A3N3[nq_, Q2_, fi_] := tA3[nq, Q2, fi] + tk1f[nq, 3]*tA4[nq, Q2, fi]
 
A2N3[nq_, Q2_, fi_] := tA2[nq, Q2, fi] + tk1f[nq, 2]*tA3[nq, Q2, fi] + 
     tk2f[nq, 2]*tA4[nq, Q2, fi]
 
A1N3[nq_, Q2_, fi_] := A1[nq, Q2, fi]
 
AN2[nq_, M_, nupp_, Q2_, fi_] := A2dN2[nq, M, nupp, Q2, fi]
 
A2dN2[nq_, 2, nupp_, Q2_, fi_] := tA2d[nq, 2 + nupp, Q2, fi]
 
A2dN2[nq_, 1, nupp_, Q2_, fi_] := tA2d[nq, 1 + nupp, Q2, fi] + 
     tk1f[nq, 1 + nupp]*tA2d[nq, 2 + nupp, Q2, fi]
 
A2dN2[nq_, 0, nupp_, Q2_, fi_] := tA2d[nq, nupp, Q2, fi] + 
     tk1f[nq, nupp]*tA2d[nq, 1 + nupp, Q2, fi] + 
     tk2f[nq, nupp]*tA2d[nq, 2 + nupp, Q2, fi]
 
A3N2[nq_, Q2_, fi_] := tA3[nq, Q2, fi]
 
A2N2[nq_, Q2_, fi_] := tA2[nq, Q2, fi] + tk1f[nq, 2]*tA3[nq, Q2, fi]
 
A1N2[nq_, Q2_, fi_] := A1[nq, Q2, fi]
 
AN1[nq_, M_, nupp_, Q2_, fi_] := A2dN1[nq, M, nupp, Q2, fi]
 
A2dN1[nq_, 1, nupp_, Q2_, fi_] := tA2d[nq, 1 + nupp, Q2, fi]
 
A2dN1[nq_, 0, nupp_, Q2_, fi_] := tA2d[nq, nupp, Q2, fi] + 
     tk1f[nq, nupp]*tA2d[nq, 1 + nupp, Q2, fi]
 
A2N1[nq_, Q2_, fi_] := tA2[nq, Q2, fi]
 
A1N1[nq_, Q2_, fi_] := A1[nq, Q2, fi]
 
rtaudBG22 = 0.19014377036321373
 
rtaudBG22 /: rtaudBG22::usage = "rtaudBG22=r_{tau}(D=0) calculated in the \
considered 2danQCD with the diagonal [2/2] Baker-Gammel approach."
 
rtauTPS = 0.17736413299472914
 
rtauTPS /: rtauTPS::usage = "rtauTPS=r_{tau}(D=0) calculated in the \
considered 2danQCD with the truncated perturbation series approach."
 
rtauLBbLB = 0.20101748748962212
 
rtauLBbLB /: rtauLBbLB::usage = "rtauLBbLB=r_{tau}(D=0) calculated (in the \
considered Lambert renormalization scheme) in the considered 2danQCD with the \
'leading-beta_0 (LB) + beyond-the leading-beta_0 (bLB)' approach with Nf=3. \
Note that here rtauLBbLB=0.2010, or 0.2030, or 0.1990"
 
d1bar = 1/12 + (9*(11/2 - 4*Zeta[3]))/4
 
d2bar = -12521/288 + (81*(302/9 - (76*Zeta[3])/3))/16 + 13*Zeta[3] + 
     (3*(401 - 424*Zeta[3] + 200*Zeta[5]))/16
 
d3bar = 144939499/20736 - (5693495*Zeta[3])/864 + (5445*Zeta[3]^2)/8 + 
     9*(1045381/15552 - (40655*Zeta[3])/864 + (5*Zeta[3]^2)/6 - 
       (260*Zeta[5])/27) + 27*(-6131/5832 + (203*Zeta[3])/324 + 
       (5*Zeta[5])/18) + (65945*Zeta[5])/288 - (7315*Zeta[7])/48 + 
     3*(-13044007/10368 + (12205*Zeta[3])/12 - 55*Zeta[3]^2 + 
       (29675*Zeta[5])/432 + (665*Zeta[7])/72)
 
td1bar = 1/12 + (9*(11/2 - 4*Zeta[3]))/4
 
td2bar = -12521/288 - (16*(1/12 + (9*(11/2 - 4*Zeta[3]))/4))/9 + 
     (81*(302/9 - (76*Zeta[3])/3))/16 + 13*Zeta[3] + 
     (3*(401 - 424*Zeta[3] + 200*Zeta[5]))/16
 
td3bar = 144939499/20736 + (8891*(1/12 + (9*(11/2 - 4*Zeta[3]))/4))/2592 - 
     (5693495*Zeta[3])/864 + (5445*Zeta[3]^2)/8 + 
     9*(1045381/15552 - (40655*Zeta[3])/864 + (5*Zeta[3]^2)/6 - 
       (260*Zeta[5])/27) + 27*(-6131/5832 + (203*Zeta[3])/324 + 
       (5*Zeta[5])/18) + (65945*Zeta[5])/288 - 
     (40*(-12521/288 + (81*(302/9 - (76*Zeta[3])/3))/16 + 13*Zeta[3] + 
        (3*(401 - 424*Zeta[3] + 200*Zeta[5]))/16))/9 - (7315*Zeta[7])/48 + 
     3*(-13044007/10368 + (12205*Zeta[3])/12 - 55*Zeta[3]^2 + 
       (29675*Zeta[5])/432 + (665*Zeta[7])/72)
 
d1 = 1.6398212048969851
 
d2 = 15.742079297915776
 
d3 = 83.55174899975434
 
td1 = 1.6398212048969851
 
td2 = 12.826841600321135
 
td3 = 34.57868801343341
 
td1bLB = 0.08333333333333326
 
td2bLB = -2.8843605694609113
 
td3bLB = 9.746656167968794
