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A340578 Decimal expansion of Product_{primes p == 5 (mod 6)} 1/(1+1/p^2).

%I A340578 #10 Jan 15 2021 06:04:51
%S A340578 9,4,4,5,0,0,9,3,4,5,0,4,7,0,0,9,8,6,7,3,4,2,9,1,0,9,4,1,9,1,4,4,4,3,
%T A340578 4,2,5,4,6,1,1,0,7,8,0,8,6,9,0,6,6,7,6,9,5,5,7,3,5,7,7,1,1,1,8,3,8,2,
%U A340578 6,4,5,1,9,9,3,3,5,7,4,6,3,9,5,6,7,7,5,3,9,6,1,7,0,5,2,9,9,4,5,3,5,8,6,7,8
%N A340578 Decimal expansion of Product_{primes p == 5 (mod 6)} 1/(1+1/p^2).
%e A340578 0.94450093450470098673429109419144434254611078086906676955735771...
%t A340578 digits = 105;
%t A340578 precision = digits + 5;
%t A340578 prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
%t A340578 Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;
%t A340578 Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
%t A340578 Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
%t A340578 gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
%t A340578 pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;
%t A340578 pD = (45*pB*Lv[2])/(4*Pi^2);
%t A340578 RealDigits[pD, 10, digits][[1]] (* Most of this code is due to _Artur Jasinski_ *)
%t A340578 (* -------------------------------------------------------------------------- *)
%t A340578 S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
%t A340578 P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
%t A340578 Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
%t A340578 $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[6, 5, 4]/Z[6, 5, 2], digits]], 10, digits-1][[1]] (* _Vaclav Kotesovec_, Jan 15 2021 *)
%Y A340578 Similar constants: A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577.
%K A340578 nonn,cons
%O A340578 0,1
%A A340578 _Jean-François Alcover_, Jan 12 2021