This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343624 #9 Apr 24 2021 03:12:19
%S A343624 0,0,0,4,6,1,3,1,5,0,5,5,3,4,3,3,8,6,9,4,0,1,7,4,5,3,0,3,3,3,4,0,9,4,
%T A343624 5,4,3,3,9,9,3,9,0,1,8,3,5,3,8,1,6,8,7,0,3,6,7,9,6,6,8,3,7,5,9,6,2,4,
%U A343624 8,9,7,8,8,5,3,2,7,9,5,2,8,8,5,0,0,2,1,9,0,0,8,5,6,6,6,8,3,6,9,7
%N A343624 Decimal expansion of the Prime Zeta modulo function P_{3,1}(4) = Sum 1/p^4 over primes p == 1 (mod 3).
%C A343624 The Prime Zeta modulo function at 4 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^4 = 1/7^4 + 1/13^4 + 1/19^4 + 1/31^4 + ...
%C A343624 The complementary Sum_{primes in A003627} 1/p^4 is given by P_{3,2}(4) = A085964 - 1/3^4 - (this value here) = 0.064186145696557778990099... = A343614.
%H A343624 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 A343624 <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%e A343624 P_{3,1}(4) = 0.000461315055343386940174530333409454339939018353816870...
%t A343624 With[{s=4}, 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 A343624 (PARI) s=0; forprimestep(p=1, 1e8, 3, s+=1./p^4); s \\ Naïve, for illustration: primes up to 10^N give about 3N+2 (= 26 for N=8) correct digits.
%o A343624 (PARI) A343606_upto(N=100)={localprec(N+5);digits((PrimeZeta31(4)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31
%Y A343624 Cf. A175645, A343625 - A343629 (P_{3,1}(3..9): same for 1/p^s, s=3, 5,..., 9).
%Y A343624 Cf. A343614 (P_{3,2}(4): same for p==2 (mod 3)), A086034 (P_{4,1}(4): same for p==1 (mod 4)), A085964 (PrimeZeta(4)).
%K A343624 cons,nonn
%O A343624 0,4
%A A343624 _M. F. Hasler_, Apr 23 2021