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
Examples
1.151925470544491047...
Links
- 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.
Programs
-
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
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
Extensions
Formula updated by Jean-François Alcover, Apr 01 2015