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

Author

Stuart Clary, Apr 15 2007

Keywords

• nonn
• cons
• easy

Comments

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

Examples

			L(3, chi3) = 0.8840238117500798567430579168710118077...
		

References

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

Links

• Index entries for transcendental numbers

Crossrefs

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

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

Cf. A344778, A344727, A344688.

Programs

Mathematica

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

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