cp's OEIS Frontend

A355234 Decimal expansion of Li_2(-1/2), the dilogarithm of (-1/2) (negated).

%I A355234 #21 Aug 06 2024 05:42:15
%S A355234 4,4,8,4,1,4,2,0,6,9,2,3,6,4,6,2,0,2,4,4,3,0,6,4,4,0,5,9,1,5,7,7,4,3,
%T A355234 2,0,8,3,4,2,6,9,9,4,1,3,4,9,1,9,9,1,2,8,5,0,1,7,4,6,3,7,1,3,1,6,8,2,
%U A355234 4,3,7,2,2,5,5,7,2,0,3,1,2,3,8,9,8,6,5,1,6,5,1,8,6,6,5,3,3,1,0,6,6,9,0,2,8
%N A355234 Decimal expansion of Li_2(-1/2), the dilogarithm of (-1/2) (negated).
%H A355234 Michael Ian Shamos, <a href="https://citeseerx.ist.psu.edu/pdf/ae33a269baba5e8b1038e719fb3209e8a00abec5">Shamos's Catalog of the Real Numbers</a>, 2011, p. 456.
%H A355234 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Dilogarithm.html">Dilogarithm</a>, eq. (26).
%H A355234 Wikipedia, <a href="https://en.wikipedia.org/wiki/Spence%27s_function">Spence's function</a>.
%F A355234 From Shamos (2011):
%F A355234 Equals -Li_2(1/3) - log(3/2)^2/2.
%F A355234 Equals Li_2(2/3) + log(3)^2/2 - log(2)^2/2 - Pi^2/6.
%F A355234 Equals Li_2(1/4)/2 + log(2)^2/2 - Pi^2/12.
%F A355234 Equals -Sum_{k>=1} (-1)^(k+1)/(2^k*k^2) = -Sum_{k>=1} (-1)^(k+1)/A007758(k).
%F A355234 Equals -Sum_{k>=1} H(k)/(k*3^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
%F A355234 Equals -Integral_{x=0..1} log(x)^2/(x+2)^2 dx.
%F A355234 Equals -Integral_{x>=1} log(x)^2/(2*x+1)^2 dx.
%F A355234 Equals Integral_{x=0..1} log(x)/(x+2) dx.
%F A355234 Equals -Integral_{x>=0} log(1 + exp(-x)/2) dx.
%e A355234 -0.44841420692364620244306440591577432083426994134919...
%t A355234 RealDigits[PolyLog[2, -1/2], 10, 100][[1]]
%o A355234 (PARI) -dilog(-1/2) \\ _Michel Marcus_, Jun 25 2022
%Y A355234 Cf. A001008, A002805, A007758.
%Y A355234 Other values of Li_2: A072691, A076788, A152115, A242599, A242600.
%K A355234 nonn,cons
%O A355234 0,1
%A A355234 _Amiram Eldar_, Jun 25 2022