%I #15 Aug 25 2021 15:46:46
%S 2,7,4,7,0,5,2,0,8,2,8,5,5,1,8,4,8,6,2,8,9,5,7,7,8,9,8,1,6,0,8,6,0,6,
%T 3,0,0,9,9,9,4,5,0,9,8,8,2,2,2,4,7,6,4,9,4,7,8,7,3,0,6,8,6,7,4,7,5,1,
%U 7,1,8,1,7,6,9,7,1,2,9,0,5,3,5,9,9,5,8,0,8,5,2,5,4,9,6,1,6,3,0,7,9,2,2,0,5,4
%N Decimal expansion of the negated Dirichlet Prime L-function of the non-principal character mod 3 at 2.
%C The absolute value of S(2,chi_2) = sum_{primes p = A000040} A102283(p)/p^2 = -1/2^2 -1/5^2 +1/7^2 -1/11^2 +1/13^2 -1/17^2 +...
%H R. J. Mathar, <a href="http://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.
%e S(2,chi_2) = -0.274705208285518486289577898160860630099...
%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); $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[-S[3, 2, 2], digits]], 10, digits-1][[1]] (* _Vaclav Kotesovec_, Jan 22 2021 *)
%Y Cf. A086241.
%K cons,nonn
%O 0,1
%A _R. J. Mathar_, Aug 01 2010
%E More digits from _Vaclav Kotesovec_, Jun 27 2020