A153066 Continued fraction for L(2, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

0, 1, 3, 1, 1, 2, 1, 17, 1, 10, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 4, 1, 1, 1, 10, 1, 2, 1, 1, 1, 6, 1, 12, 2, 14, 1, 1, 1, 3, 3, 1, 1, 3, 1, 1, 12, 3, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 4, 2, 1, 12, 140, 1, 6, 3, 3, 1, 2, 1100, 4, 1, 1, 2, 1

Offset: 0

Author

Stuart Clary, Dec 17 2008

Keywords

• nonn
• cofr
• easy

Examples

			L(2, chi3) = 0.781302412896486296867187429624092... = A086724 = = [0; 1, 3, 1, 1, 2, 1, 17, 1, 10, 1, 1, 5, 1, 1, 2, 1, ...]
		

Crossrefs

Cf. A153067, A153068.

Programs

Mathematica

nmax = 1000; ContinuedFraction[(Zeta[2, 1/3] - Zeta[2, 2/3])/9, nmax + 1]

PARI

contfrac(zetahurwitz(2,1/3)/9 - zetahurwitz(2,2/3)/9) \\ Charles R Greathouse IV, Jan 31 2018

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.

Series: L(2, chi3) = sum_{k=1..infinity} chi3(k) k^{-2} = 1 - 1/2^2 + 1/4^2 - 1/5^2 + 1/7^2 - 1/8^2 + 1/10^2 - 1/11^2 + ...