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

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, 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, 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

Offset: 0

Author

Amiram Eldar, Jun 25 2022

Keywords

• nonn
• cons

Examples

			-0.44841420692364620244306440591577432083426994134919...
		

Links

• Michael Ian Shamos, Shamos's Catalog of the Real Numbers, 2011, p. 456.
• Eric Weisstein's World of Mathematics, Dilogarithm, eq. (26).
• Wikipedia, Spence's function.

Crossrefs

Cf. A001008, A002805, A007758.

Other values of Li_2: A072691, A076788, A152115, A242599, A242600.

Programs

Mathematica

RealDigits[PolyLog[2, -1/2], 10, 100][[1]]

PARI

-dilog(-1/2) \\ Michel Marcus, Jun 25 2022

Formula

From Shamos (2011):

Equals -Li_2(1/3) - log(3/2)^2/2.

Equals Li_2(2/3) + log(3)^2/2 - log(2)^2/2 - Pi^2/6.

Equals Li_2(1/4)/2 + log(2)^2/2 - Pi^2/12.

Equals -Sum_{k>=1} (-1)^(k+1)/(2^k*k^2) = -Sum_{k>=1} (-1)^(k+1)/A007758(k).

Equals -Sum_{k>=1} H(k)/(k*3^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Equals -Integral_{x=0..1} log(x)^2/(x+2)^2 dx.

Equals -Integral_{x>=1} log(x)^2/(2*x+1)^2 dx.

Equals Integral_{x=0..1} log(x)/(x+2) dx.

Equals -Integral_{x>=0} log(1 + exp(-x)/2) dx.