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%I #35 Feb 04 2021 05:43:10
%S 1,0,6,0,5,4,8,2,9,3,1,6,9,1,1,0,7,2,8,1,7,4,1,2,6,3,6,4,3,0,9,8,7,2,
%T 0,3,4,9,3,0,7,7,1,3,0,2,0,4,4,8,7,1,6,3,1,2,7,9,9,4,3,7,2,1,8,1,7,9,
%U 4,6,0,8,0,2,4,4,0,6,6,3,7,4,5,9,0,3,1,6,1,4,3,8,7,6,8,5,6,3,3,5,6,5,0,1,5
%N Decimal expansion of Product_{primes p == 5 (mod 6)} 1/(1-1/p^2).
%C The four similar sequences for products of primes mod 6 are these:
%C A175646 for primes p == 1 (mod 6)} 1/(1-1/p^2),
%C A340576 for primes p == 5 (mod 6)} 1/(1-1/p^2),
%C A340577 for primes p == 1 (mod 6)} 1/(1+1/p^2),
%C A340578 for primes p == 5 (mod 6)} 1/(1+1/p^2).
%H R. J. Mathar, <a href="https://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and prime zeta modulo functions for small moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015, Zeta_{6,5}(2) in section 3.2.
%F g = A143298 = (9 - PolyGamma(1, 2/3) + PolyGamma(1, 4/3))/(4 sqrt(3));
%F h = A301429;
%F Equals (3*sqrt(3)*h^2)/2.
%F Equals (3/4)*A333240.
%F A340577 = Pi^4/(243*g*h^2);
%F A340578 = (45*g*h^2)/(2*Pi^2).
%F Equals Pi^2/(9*A175646). - _Artur Jasinski_, Jan 11 2021
%F Equals Sum_{k>=1} 1/A259548(k)^2. - _Amiram Eldar_, Jan 24 2021
%e 1.06054829316911072817412636430987203493077130204487163127994372...
%p a := n -> 3^(2^(-n-2))*((1-3^(-2^(n+1)))/2)^(2^(-n-1)):
%p b := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
%p c := n -> a(n)*b(2^(n+1))^(1/2^(n+1)):
%p Digits := 107: evalf((3/4)*mul(c(n), n=0..9)); # _Peter Luschny_, Jan 14 2021
%t digits = 105;
%t precision = digits + 10;
%t prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
%t Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision] &;
%t Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
%t Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
%t gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
%t pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;
%t RealDigits[pB, 10, digits][[1]] (* Most of this code is due to _Artur Jasinski_ *)
%t 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 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 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 $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[6,5,2], digits]], 10, digits-1][[1]] (* _Vaclav Kotesovec_, Jan 15 2021 *)
%Y Similar constants: A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340577, A340578.
%Y Cf. A259548.
%K nonn,cons
%O 1,3
%A _Jean-François Alcover_, Jan 12 2021
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