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A261623 Decimal expansion of the Dirichlet beta function at 1/4.

%I A261623 #15 Feb 16 2025 08:33:26
%S A261623 5,9,0,7,2,3,0,5,6,4,4,2,4,9,4,7,3,1,8,6,5,9,5,9,1,5,3,5,1,1,5,6,2,0,
%T A261623 5,9,7,9,8,3,6,7,4,1,7,2,3,9,1,1,4,4,0,0,8,2,7,7,1,8,7,6,5,9,3,0,0,5,
%U A261623 8,3,1,8,2,0,6,6,4,5,9,6,0,9,6,9,2,8,7,7,2,6,1,3,4,1,4,2,0,1,1,7,3,9,4
%N A261623 Decimal expansion of the Dirichlet beta function at 1/4.
%H A261623 G. C. Greubel, <a href="/A261623/b261623.txt">Table of n, a(n) for n = 0..10000</a>
%H A261623 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/DirichletBetaFunction.html">Dirichlet Beta Function</a>
%H A261623 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_beta_function">Dirichlet beta function</a>
%F A261623 beta(1/4) = (zeta(1/4, 1/4) - zeta(1/4, 3/4))/sqrt(2).
%e A261623 0.59072305644249473186595915351156205979836741723911440082771876593...
%p A261623 evalf(Sum((-1)^n/(2*n+1)^(1/4), n=0..infinity), 120); # _Vaclav Kotesovec_, Aug 27 2015
%t A261623 RealDigits[DirichletBeta[1/4],10,103]//First
%o A261623 (PARI) beta(x)=(zetahurwitz(x, 1/4)-zetahurwitz(x, 3/4))/4^x
%o A261623 beta(1/4) \\ _Charles R Greathouse IV_, Oct 18 2024
%Y A261623 Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A261622 (beta(1/3)), A261624 (beta(1/5)).
%K A261623 nonn,cons,easy
%O A261623 0,1
%A A261623 _Jean-François Alcover_, Aug 27 2015