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Decimal expansion of the negated Dirichlet Prime L-function of the real non-principal character mod 5 at 1.
1

%I #13 Aug 25 2021 13:40:42

%S 1,0,0,7,9,9,6,5,4,7,9,3,9,8,6,1,1,7,2,2,6,1,6,6,6,0,7,5,5,1,2,6,7,8,

%T 5,6,6,9,9,9,0,3,1,9,5,6,6,4,9,3,2,7,0,9,7,1,6,3,7,4,5,5,3,9,5,0,0,1,

%U 4,9,4,5,8,8,5,3,9,3,2,4,8,6,4,3,3,8,6,8,1,3,3,8,6,3,3,7,3,8,2,7,2,3,7,6,2

%N Decimal expansion of the negated Dirichlet Prime L-function of the real non-principal character mod 5 at 1.

%C The absolute value of S(1,chi_3) = sum_{primes p = A000040} A080891(p)/p = -1/2 -1/3 -1/7 +1/11 -1/13 -1/17 -1/23 +...

%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(1,chi_3) = -1.0079965479398611722616660755126785669990319566493...

%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[5, 3, 1], digits]], 10, digits-1][[1]] (* _Vaclav Kotesovec_, Jan 22 2021 *)

%Y Cf. A086241 (mod 3), A086239 (mod 4), A175643 (mod 6).

%K cons,nonn

%O 1,4

%A _R. J. Mathar_, Aug 01 2010

%E More digits from _Vaclav Kotesovec_, Jan 22 2021