  We are interested in finding for which positive integers $D$ we have rational
solutions for the equation $x^3+y^3=D.$ 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 $p\equiv 1\mod 9$, 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 $\mathbb{Q}[\sqrt{-3}]$.
