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%I #8 Apr 24 2021 03:12:44
%S 0,0,0,0,6,2,6,5,5,4,2,7,4,7,1,7,5,5,5,0,6,0,0,2,5,6,9,1,9,1,0,2,4,0,
%T 8,8,4,4,6,4,7,5,7,2,0,6,7,2,6,2,0,8,2,4,1,0,6,9,5,1,6,1,4,3,6,3,6,9,
%U 7,5,1,8,8,8,4,1,3,4,3,0,7,9,7,0,3,6,1,4,6,9,3,7,9,9,5,1,9,7,3,3
%N Decimal expansion of the Prime Zeta modulo function P_{3,1}(5) = Sum 1/p^5 over primes p == 1 (mod 3).
%C The Prime Zeta modulo function at 5 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^5 = 1/7^5 + 1/13^5 + 1/19^5 + 1/31^5 + ...
%C The complementary Sum_{primes in A003627} 1/p^5 is given by P_{3,2}(5) = A085965 - 1/3^5 - (this value here) = 0.03157713571900394195603378... = A343615.
%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, p.21.
%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%e P_{3,1}(5) = 6.2655427471755506002569191024088446475720672620824106951614...*10^-5
%t With[{s=5}, Do[Print[N[1/2 * Sum[(MoebiusMu[2*n + 1]/(2*n + 1)) * Log[(Zeta[s + 2*n*s]*(Zeta[s + 2*n*s, 1/6] - Zeta[s + 2*n*s, 5/6])) / ((1 + 2^(s + 2*n*s))*(1 + 3^(s + 2*n*s)) * Zeta[2*(1 + 2*n)*s])], {n, 0, m}], 120]], {m, 100, 500, 100}]] (* adopted from _Vaclav Kotesovec_'s code in A175645 *)
%o (PARI) s=0; forprimestep(p=1, 1e8, 3, s+=1./p^5); s \\ Naïve, for illustration: primes up to 10^N give 4N+2 (= 34 for N=8) correct digits.
%o (PARI) A343607_upto(N=100)={localprec(N+5);digits((PrimeZeta31(5)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31
%Y Cf. A086035 (P_{4,1}(5): same for p==1 (mod 4)), A175645, A343624 - A343629 (P_{3,1}(3..9): same for 1/p^s, s=3..9), A343615 (P_{3,2}(5): same for p==2 (mod 3)).
%Y Cf. A085965 (PrimeZeta(5)), A002476 (primes of the form 3k+1).
%K cons,nonn
%O 0,5
%A _M. F. Hasler_, Apr 23 2021