cp's OEIS Frontend

A175571 Decimal expansion of the Dirichlet beta function of 5.

%I A175571 #33 Feb 16 2025 08:33:12
%S A175571 9,9,6,1,5,7,8,2,8,0,7,7,0,8,8,0,6,4,0,0,6,3,1,9,3,6,8,6,3,0,9,7,5,2,
%T A175571 8,1,5,1,1,3,9,5,5,2,9,3,8,8,2,6,4,9,4,3,2,0,7,9,8,3,2,1,5,1,2,4,4,6,
%U A175571 2,8,6,5,0,1,8,2,7,4,8,1,9,2,8,9,6,5,9,8,3,2,2,7,0,5,2,4,4,7,5,5,9,9,0,8,0
%N A175571 Decimal expansion of the Dirichlet beta function of 5.
%C A175571 The value of the Dirichlet L-series L(m=4,r=2,s=4), see arXiv:1008.2547.
%D A175571 L. B. W. Jolley, Summation of Series, Dover, 1961, eq. 308.
%H A175571 G. C. Greubel, <a href="/A175571/b175571.txt">Table of n, a(n) for n = 0..10000</a>
%H A175571 Richard J. Mathar, <a href="https://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.
%H A175571 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletBetaFunction.html">Dirichlet Beta Function</a>.
%H A175571 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_beta_function">Dirichlet beta function</a>.
%F A175571 Equals 5*Pi^5/1536 = Sum_{n>=1} A101455(n)/n^5, where Pi^5 = A092731. [corrected by _R. J. Mathar_, Feb 01 2018]
%F A175571 Equals Sum_{n>=0} (-1)^n/(2*n+1)^5. - _Jean-François Alcover_, Mar 29 2013
%F A175571 Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^5)^(-1). - _Amiram Eldar_, Nov 06 2023
%e A175571 0.99615782807708806400631936...
%p A175571 DirichletBeta := proc(s) 4^(-s)*(Zeta(0,s,1/4)-Zeta(0,s,3/4)) ; end proc: x := DirichletBeta(5) ; x := evalf(x) ;
%t A175571 RealDigits[ DirichletBeta[5], 10, 105] // First (* _Jean-François Alcover_, Feb 20 2013, updated Mar 14 2018 *)
%o A175571 (PARI) 5*Pi^5/1536 \\ _Charles R Greathouse IV_, Jan 31 2018
%o A175571 (PARI) beta(x)=(zetahurwitz(x,1/4)-zetahurwitz(x,3/4))/4^x
%o A175571 beta(5) \\ _Charles R Greathouse IV_, Jan 31 2018
%Y A175571 Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).
%Y A175571 Cf. A101455.
%K A175571 cons,easy,nonn
%O A175571 0,1
%A A175571 _R. J. Mathar_, Jul 15 2010