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 A175571 Decimal expansion of the Dirichlet beta function of 5. 14
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The value of the Dirichlet L-series L(m=4,r=2,s=4), see arXiv:1008.2547. REFERENCES L. B. W. Jolley, Summation of Series, Dover, 1961, eq. 308. LINKS G. C. Greubel, Table of n, a(n) for n = 0..10000 Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015. Eric Weisstein's World of Mathematics, Dirichlet Beta Function. Wikipedia, Dirichlet beta function. FORMULA Equals 5*Pi^5/1536 = Sum_{n>=1} A101455(n)/n^5, where Pi^5 = A092731. [corrected by R. J. Mathar, Feb 01 2018] Equals Sum_{n>=0} (-1)^n/(2*n+1)^5. - Jean-François Alcover, Mar 29 2013 Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^5)^(-1). - Amiram Eldar, Nov 06 2023 EXAMPLE 0.99615782807708806400631936... MAPLE DirichletBeta := proc(s) 4^(-s)*(Zeta(0, s, 1/4)-Zeta(0, s, 3/4)) ; end proc: x := DirichletBeta(5) ; x := evalf(x) ; MATHEMATICA RealDigits[ DirichletBeta[5], 10, 105] // First (* Jean-François Alcover, Feb 20 2013, updated Mar 14 2018 *) PROG (PARI) 5*Pi^5/1536 \\ Charles R Greathouse IV, Jan 31 2018 (PARI) beta(x)=(zetahurwitz(x, 1/4)-zetahurwitz(x, 3/4))/4^x beta(5) \\ Charles R Greathouse IV, Jan 31 2018 CROSSREFS 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)). Cf. A101455. Sequence in context: A340222 A021505 A346930 * A019894 A347150 A346850 Adjacent sequences: A175568 A175569 A175570 * A175572 A175573 A175574 KEYWORD cons,easy,nonn AUTHOR R. J. Mathar, Jul 15 2010 STATUS approved

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Last modified February 21 03:13 EST 2024. Contains 370219 sequences. (Running on oeis4.)