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A175571 Decimal expansion of the Dirichlet beta function of 5. 10
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

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]

Also equals Sum_{n>=0} (-1)^n/(2*n+1)^5. - Jean-François Alcover, Mar 29 2013

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 (at 1), A006752 (at 2), A153071 (at 3), A175572 (at 4), A175570 (at 6).

Sequence in context: A144668 A340222 A021505 * A019894 A344505 A334478

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 August 1 07:45 EDT 2021. Contains 346384 sequences. (Running on oeis4.)