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%I #39 Dec 17 2023 03:05:59
%S 9,4,9,7,0,3,1,2,6,2,9,4,0,0,9,3,9,5,2,6,3,4,9,8,4,9,1,7,4,5,7,4,1,5,
%T 1,5,8,7,3,6,5,1,9,5,0,9,0,9,6,9,2,9,4,4,8,8,0,9,1,7,6,5,4,3,6,8,3,0,
%U 5,1,9,5,5,6,8,7,9,2,8,8,1,7,2,6,0,0,6,8,0,3,2,8,4,8,3,5,3,5,0,1,6,8,7,2,9,0
%N Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^(24*k)) dx.
%C Integral_{x=-1..1} Product_{k>=1} (1-x^(24*k)) dx = Pi^2/(3*sqrt(3)) = 1.89940625258801878... . - _Vaclav Kotesovec_, Jun 02 2015
%C Equals the value of the Dirichlet L-series of a non-principal character modulo 12 (A110161) at s=2. - _Jianing Song_, Nov 16 2019
%H M. W. Coffey, <a href="https://arxiv.org/abs/1701.07064">Summatory relations and prime products for the Stieltjes constants and other related results</a>, arXiv:1701.07064 [math.NT], 2017, eq. (2.1).
%H Vaclav Kotesovec, <a href="http://oeis.org/A258232/a258232_2.pdf">The integration of q-series</a>.
%F Equals Pi^2/(6*sqrt(3)).
%F Equals Sum_{k>=1} A110161(n)/k^2 = Sum_{k>=1} Kronecker(12,k)/k^2. - _Jianing Song_, Nov 16 2019
%F Equals -Integral_{x=0..oo} log(x)/(x^6 + 1) dx. - _Amiram Eldar_, Aug 12 2020
%F Equals 1 + Sum_{k>=1} ( (-1)^k/(6*k-1)^2 + (-1)^k/(6*k+1)^2 ). - _Sean A. Irvine_, Jul 18 2021
%F Equals 1/(Product_{p prime == 1 or 11 (mod 12)} (1 - 1/p^2) * Product_{p prime == 5 or 7 (mod 12)} (1 + 1/p^2)). - _Amiram Eldar_, Dec 17 2023
%e 0.9497031262940093952634984917457415158736519509096929448809176543683...
%p evalf(Pi^2/(6*sqrt(3)), 120);
%t RealDigits[Pi^2/(6*Sqrt[3]),10,120][[1]]
%t N[Sum[(-1)^n/(12*n*(3n-1)+1),{n,-Infinity,Infinity}],105]
%Y Cf. A258232, A258406, A258408.
%Y Cf. A196530, A110161.
%Y Cf. A346451, A346450, A346449.
%K nonn,cons,easy
%O 0,1
%A _Vaclav Kotesovec_, May 29 2015
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