A129404 Decimal expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.
8, 8, 4, 0, 2, 3, 8, 1, 1, 7, 5, 0, 0, 7, 9, 8, 5, 6, 7, 4, 3, 0, 5, 7, 9, 1, 6, 8, 7, 1, 0, 1, 1, 8, 0, 7, 7, 4, 7, 9, 4, 6, 1, 8, 6, 1, 1, 7, 6, 5, 8, 9, 3, 4, 7, 8, 2, 5, 8, 7, 4, 1, 4, 7, 4, 9, 1, 1, 5, 6, 6, 7, 0, 3, 3, 3, 2, 3, 1, 8, 7, 0, 1, 6, 3, 5, 9, 6, 3, 6, 4, 6, 8, 9, 5, 5, 3, 6, 0, 6
Offset: 0
Examples
L(3, chi3) = 0.8840238117500798567430579168710118077...
References
- Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292.
Crossrefs
Programs
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Mathematica
nmax = 1000; First[ RealDigits[4 Pi^3/(81 Sqrt[3]) - (1/2) * 10^(-nmax), 10, nmax] ]
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PARI
4*Pi^3/81/sqrt(3) \\ Charles R Greathouse IV, Sep 02 2024
Formula
chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A102283 (A049347 shifted).
Series: L(3, chi3) = Sum_{k>=1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
Equals 1 + Sum_{k>=1} ( 1/(3*k+1)^3 - 1/(3*k-1)^3 ). - Sean A. Irvine, Aug 17 2021
Equals Product_{p prime} (1 - Kronecker(-3, p)/p^3)^(-1) = Product_{p prime != 3} (1 + (-1)^(p mod 3)/p^3)^(-1). - Amiram Eldar, Nov 06 2023
Extensions
Offset corrected by R. J. Mathar, Feb 05 2009
Comments