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%I #38 Nov 07 2023 04:56:38
%S 9,8,8,9,4,4,5,5,1,7,4,1,1,0,5,3,3,6,1,0,8,4,2,2,6,3,3,2,2,8,3,7,7,8,
%T 2,1,3,1,5,8,6,0,8,8,7,0,6,2,7,3,3,9,1,0,7,8,1,9,9,2,4,0,1,6,3,9,0,1,
%U 5,1,9,4,6,9,8,0,1,8,1,9,6,4,1,1,9,1,0,4,6,8,9,9,9,7,9,9,9,3,3,7,8,5,6,2,1
%N Decimal expansion of the Dirichlet beta function of 4.
%C This is the value of the Dirichlet L-series for A101455 at s=4, see arXiv:1008.2547, L(m=4,r=2,s=4).
%D L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (308).
%H G. C. Greubel, <a href="/A175572/b175572.txt">Table of n, a(n) for n = 0..10000</a>
%H Mark 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, proposition 5.
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletBetaFunction.html">Dirichlet Beta Function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_beta_function">Dirichlet beta function</a>.
%F Equals Sum_{n>=1} A101455(n)/n^4. [corrected by _R. J. Mathar_, Feb 01 2018]
%F Equals (PolyGamma(3, 1/4) - PolyGamma(3, 3/4))/1536. - _Jean-François Alcover_, Jun 11 2015
%F Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^4)^(-1). - _Amiram Eldar_, Nov 06 2023
%e 0.988944551741105336108422633...
%p DirichletBeta := proc(s) 4^(-s)*(Zeta(0,s,1/4)-Zeta(0,s,3/4)) ; end proc: x := DirichletBeta(4) ; x := evalf(x) ;
%t RealDigits[ DirichletBeta[4], 10, 105] // First (* _Jean-François Alcover_, Feb 11 2013, updated Mar 14 2018 *)
%o (PARI) beta(x)=(zetahurwitz(x,1/4)-zetahurwitz(x,3/4))/4^x
%o beta(4) \\ _Charles R Greathouse IV_, Jan 31 2018
%Y Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175571 (beta(5)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).
%Y Cf. A101455.
%K cons,easy,nonn
%O 0,1
%A _R. J. Mathar_, Jul 15 2010
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