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A129404 Decimal expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

%I A129404 #22 Dec 04 2024 08:16:25
%S A129404 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,
%T A129404 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,
%U A129404 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
%N A129404 Decimal expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.
%C A129404 Contributed to OEIS on Apr 15 2007 -- the 300th anniversary of the birth of Leonhard Euler.
%D A129404 Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292.
%H A129404 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A129404 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).
%F A129404 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 A129404 Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
%F A129404 Equals 1 + Sum_{k>=1} ( 1/(3*k+1)^3 - 1/(3*k-1)^3 ). - _Sean A. Irvine_, Aug 17 2021
%F A129404 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
%e A129404 L(3, chi3) = 0.8840238117500798567430579168710118077...
%t A129404 nmax = 1000; First[ RealDigits[4 Pi^3/(81 Sqrt[3]) - (1/2) * 10^(-nmax), 10, nmax] ]
%o A129404 (PARI) 4*Pi^3/81/sqrt(3) \\ _Charles R Greathouse IV_, Sep 02 2024
%Y A129404 Cf. A129405, A129406, A129407, A129408, A129409, A129410, A129411.
%Y A129404 Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665.
%Y A129404 Cf. A344778, A344727, A344688.
%K A129404 nonn,cons,easy
%O A129404 0,1
%A A129404 _Stuart Clary_, Apr 15 2007
%E A129404 Offset corrected by _R. J. Mathar_, Feb 05 2009