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

0,1

Comments

Contributed to OEIS on Apr 15 2007 -- the 300th anniversary of the birth of Leonhard Euler.

References

Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292.

Links

Table of n, a(n) for n=0..99.

Index entries for transcendental numbers

Formula

chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is 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

Example

L(3, chi3) = 0.8840238117500798567430579168710118077...

Mathematica

nmax = 1000; First[ RealDigits[4 Pi^3/(81 Sqrt[3]) - (1/2) * 10^(-nmax), 10, nmax] ]

Prog

(PARI) 4*Pi^3/81/sqrt(3) \\ Charles R Greathouse IV, Sep 02 2024

Crossrefs

Cf. A129405, A129406, A129407, A129408, A129409, A129410, A129411.

Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665.

Cf. A344778, A344727, A344688.

Sequence in context: A011213 A178728 A256489 * A222075 A117040 A085669

Adjacent sequences: A129401 A129402 A129403 * A129405 A129406 A129407

Keyword

nonn,cons,easy

Author

Stuart Clary, Apr 15 2007

Extensions

Offset corrected by R. J. Mathar, Feb 05 2009

Status

approved