This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340004 #82 Jan 31 2021 08:35:17
%S A340004 1,0,1,0,9,1,5,1,6,0,6,0,1,0,1,9,5,2,2,6,0,4,9,5,6,5,8,4,2,8,9,5,1,4,
%T A340004 9,2,0,9,8,4,5,3,8,6,2,7,5,8,1,7,3,8,5,2,3,7,3,2,0,2,4,2,0,0,8,9,2,5,
%U A340004 1,6,1,3,7,4,2,4,5,6,7,2,6,3,7,0,9,3,9,6,1,9,7,6,9,4,5,5,8,9,2,1,8
%N A340004 Decimal expansion of Product_{primes p == 1 (mod 5)} p^2/(p^2-1).
%C A340004 This constant is called Euler product 2==1 modulo 5 (see Mathar's Definition 5 formula (38)) or equivalently zeta 2==1 modulo 5.
%H A340004 Vaclav Kotesovec, <a href="/A340004/b340004.txt">Table of n, a(n) for n = 1..501</a>
%H A340004 Salma Ettahri, Olivier Ramaré, and Léon Surel, <a href="https://arxiv.org/abs/1908.06808">Fast multi-precision computation of some Euler products</a>, arXiv:1908.06808 [math.NT], 2019 p.20 (100 digits precision data).
%H A340004 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], 2014-2015, Section 3.3. zeta_{5,1}(2).
%F A340004 Equals Sum_{k>=1} 1/A004615(k)^2. - _Amiram Eldar_, Jan 24 2021
%F A340004 Equals exp(-gamma/2)*Pi/(A340839^2*sqrt(5*log((1 + sqrt (5))/2))). - _Artur Jasinski_, Jan 30 2021
%e A340004 1.01091516060101952260495658428951492...
%t A340004 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 A340004 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 A340004 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 A340004 $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[5, 1, 2], digits]], 10, digits-1][[1]] (* _Vaclav Kotesovec_, Jan 15 2021, took 20 minutes *)
%Y A340004 Cf. A340127, A340628, A340629, A340665, A340794, A340839.
%Y A340004 Cf. A175646, A301429, A333240.
%Y A340004 Cf. A175647, A248930, A248938, A335963.
%Y A340004 Cf. A004615, A340576, A340577, A340578, A334826.
%K A340004 nonn,cons
%O A340004 1,5
%A A340004 _Artur Jasinski_, Jan 15 2021