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%I #16 May 07 2021 08:07:55
%S 1,3,4,1,2,5,1,7,8,9,1,5,4,6,3,5,4,0,4,2,8,5,9,9,3,2,9,9,9,9,4,3,1,1,
%T 9,8,9,9,5,8,7,9,9,1,9,7,5,2,1,6,8,3,3,7,3,7,0,5,9,9,1,0,6,1,5,3,8,5,
%U 3,3,4,9,9,5,6,0,4,7,9,3,7,6,7,1,5,2,8,6,5,3,7,4,0,4,0,3,4,4,4,3,3,6,7,8,6
%N Decimal expansion of P_{3,2}(3) = Sum 1/p^3 over primes == 2 (mod 3).
%C The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.
%H Jean-François Alcover, <a href="/A343613/b343613.txt">Table of n, a(n) for n = 0..1006</a>
%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, value P(m=3, n=2, s=3) on p. 21.
%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%F P_{3,2}(3) = P(3) - 1/3^3 - P_{3,1}(3) = A085541 - A021031 - A175645.
%e 0.134125178915463540428599329999431198995879919752168337370599106153853349956...
%o (PARI) s=0;forprimestep(p=2,1e8,3,s+=1./p^3);s \\ For illustration: using primes up to 10^N gives about 2N+2 (= 18 for N=8) correct digits.
%o (PARI) A343613_upto(N=100)={localprec(N+5); digits((PrimeZeta32(3)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32.
%Y Cf. A003627 (primes 3k-1), A085541 (PrimeZeta(3)), A021031 (1/27).
%Y Cf. A175645 (same for p==1 (mod 3)), A086033 (for primes 4k+1), A085992 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).
%K nonn,cons
%O 0,2
%A _M. F. Hasler_, Apr 22 2021
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