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Expansion of L(3, chi3) in base 3, where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.
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%I #18 Sep 02 2024 01:28:47

%S 2,1,2,2,1,2,1,1,0,0,2,0,1,1,1,1,1,0,1,0,2,2,0,2,2,0,2,0,0,0,2,1,0,2,

%T 2,1,1,0,0,1,1,1,0,2,1,1,1,0,0,2,2,0,0,0,0,2,0,2,2,0,0,0,0,2,2,0,0,1,

%U 0,2,1,1,2,2,2,0,0,2,2,1,0,2,0,1,2,2,2,1,2,1,1,1,1,0,2,2,0,2,1,1

%N Expansion of L(3, chi3) in base 3, where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

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

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

%F 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.

%F 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 + ...

%F Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).

%e L(3, chi3) = 0.8840238117500798567430579168710118077... = (0.2122121100201111101022022020002102211...)_3

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

%Y Cf. A129404, A129405, A129407, A129408, A129409, A129410, A129411.

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

%K nonn,base,cons,easy

%O 0,1

%A _Stuart Clary_, Apr 15 2007

%E Offset corrected by _R. J. Mathar_, Feb 05 2009