Central values of $L$-functions of cubic twists

We are interested in finding for which positive integers $D$ we have rational solutions for the equation $x^3+y^3=D\mathrm{.}$ The aim of this paper is to compute the value of the $L$-function $L(E_D,1)$ for the elliptic curves $E_D:x^3+y^3=D$. For the case of $p$ prime $p\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}9$, two formulas have been computed by Rodriguez-Villegas and Zagier. We have computed formulas that relate $L(E_D, 1)$ to the square of a trace of a modular function at a CM point. This offers a criterion for when the integer $D$ is the sum of two rational cubes. Furthermore, when $L(E_D,1)$ is nonzero we get a formula for the number of elements in the Tate-Shafarevich group and we show that this number is a square when $D$ is a norm in $\mathbb{Q}[\sqrt{-3}]$.