[math-fun] Mathematica knows that Hypergeometric2F1[1/3, 2/3, 1, 1/2] =

Gamma[1/3]/(2^(1/3) Gamma[2/3]^2) So it's only natural that a formula of Bruce Berndt should give Hypergeometric2F1[1/3, 2/3, 1, -1] == -(((-1)^(11/12) 3^( 3/4) (2 (674484539 - 581632 Sqrt[2])^(2/3) - 1734 (-674484539 + 581632 Sqrt[2])^(1/3) + 769097 I (I + Sqrt[3]))^(1/4) EllipticK[1/32 (16 - \[Sqrt](6 (-14867 + 17 (-674484539 - 581632 Sqrt[2])^(1/3) - 17 (-1)^(2/3) (674484539 - 581632 Sqrt[2])^( 1/3))))])/(2 Sqrt[2] (674484539 - 581632 Sqrt[2])^( 1/12) \[Pi])) --rwg (-: Explanation under construction)