A175643 Decimal expansion of the negated Dirichlet Prime L-function of the real non-principal character mod 6 at 1.

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

Offset: 0

Author

R. J. Mathar, Aug 01 2010

Keywords

• cons
• nonn

Comments

The absolute value of S(1,chi_2) = sum_{primes p = A000040} A134667(p)/p = -1/5 +1/7 -1/11+1/13 -1/17 +1/19 -1/23 +...

Examples

			S(1,chi_2) = -0.14194483853319570866139263972173431667541104401...
		

Links

• R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Crossrefs

Cf. A086241 (mod 3), A086239 (mod 4), A175642 (mod 5).

Programs

Mathematica

Do[Print[N[-Log[4/3]/2 + Sum[Log[(Zeta[2*k + 1, 1/6] - Zeta[2*k + 1, 5/6])^2 / ((2^(4*k + 2) - 1) * (3^(4*k + 2) - 1) * Zeta[4*k + 2])] * MoebiusMu[2*k + 1]/(4*k + 2), {k, 1, m}], 120]], {m, 20, 200, 20}] (* Vaclav Kotesovec, Jun 27 2020 *)
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[6, 2, 1], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 22 2021 *)

Extensions

More terms from Vaclav Kotesovec, Jun 27 2020