A103133 Decimal expansion of Dirichlet series L_{-7}(2).

1, 1, 5, 1, 9, 2, 5, 4, 7, 0, 5, 4, 4, 4, 9, 1, 0, 4, 7, 1, 0, 1, 6, 9, 2, 3, 9, 7, 3, 2, 0, 5, 4, 9, 9, 6, 4, 7, 9, 7, 8, 2, 1, 4, 0, 4, 6, 8, 6, 5, 6, 6, 9, 1, 4, 0, 8, 3, 9, 6, 8, 6, 3, 6, 1, 6, 6, 1, 2, 4, 1, 6, 3, 4, 5, 4, 5, 9, 1, 5, 4, 7, 5, 5, 6, 6, 7, 7, 5, 1, 9, 0, 6, 2, 9, 7, 2, 1, 2, 5, 3, 4

Offset

1,3

Links

Table of n, a(n) for n=1..102.

D. Cvijovic, Proof of the Borwein-Broadhurst conjecture for a dilogarithmic integral arising in quantum field theory, arXiv:1011.0195 [math-ph], 2010.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 98-99.

R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, L(m=7,r=4,s=2).

Eric Weisstein's World of Mathematics, Dirichlet L-Series.

Formula

(Psi(1, 1/7) + Psi(1, 2/7) - Psi(1, 3/7) + Psi(1, 4/7) - Psi(1, 5/7) - Psi(1, 6/7))/49, where Psi(1, x) is the polygamma function of order 1.

Equals Sum_{n>=1} A175629(n)/n^2. - R. J. Mathar, Jan 15 2021

Equals 1/(Product_{p prime == 1, 2 or 4 (mod 7)} (1 - 1/p^2) * Product_{p prime == 3, 5 or 6 (mod 7)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023

Example

1.151925470544491047...

Mathematica

(PolyGamma[1, 1/7] + PolyGamma[1, 2/7] - PolyGamma[1, 3/7] + PolyGamma[1, 4/7] - PolyGamma[1, 5/7] - PolyGamma[1, 6/7])/49 // RealDigits[#, 10, 102]& // First

Prog

(PARI) sumnumrat((564950498*n^9 + 2542277241*n^8 + 5007141440*n^7 + 5661034582*n^6 + 4045007918*n^5 + 1891913569*n^4 + 578372088*n^3 + 111310360*n^2 + 12239136*n + 587664)/(13841287201*n^12 + 83047723206*n^11 + 223437921959*n^10 + 355918813740*n^9 + 373230511143*n^8 + 270934117398*n^7 + 139314416297*n^6 + 51007900440*n^5 + 13163376856*n^4 + 2328240096*n^3 + 267062544*n^2 + 17781120*n + 518400), 0) \\ Charles R Greathouse IV, Jul 08 2026
(PARI) (zetahurwitz(2, 1/7) + zetahurwitz(2, 2/7) - zetahurwitz(2, 3/7) + zetahurwitz(2, 4/7) - zetahurwitz(2, 5/7) - zetahurwitz(2, 6/7))/49 \\ Charles R Greathouse IV, Jul 08 2026

Crossrefs

Cf. A326919, A327135, A175629.

Sequence in context: A147326 A336048 A143114 * A098318 A360750 A293198

Adjacent sequences: A103130 A103131 A103132 * A103134 A103135 A103136

Keyword

nonn,cons,changed

Author

Eric W. Weisstein, Jan 23 2005

Extensions

Formula updated by Jean-François Alcover, Apr 01 2015

Status

approved