$L$-series values for sextic twists of elliptic curves over $\mathbb{Q}[\sqrt{-3}]$

We prove a new formula for the central value of the $L$-function $L(E_{D, α}, 1)$ corresponding to the family of sextic twists over $\mathbb{Q}[\sqrt{-3}]$ of elliptic curves $E_{D, α}: y^2=x^3+16D^2α^3$ for $D$ an integer and $α\in \mathbb{Q}[\sqrt{-3}]$. The formula generalizes the result of cubic twists over $\mathbb{Q}$ of Rodriguez-Villegas and Zagier for a prime $D \equiv 1 (9)$ and of Rosu for general $D$. For $α$ prime and all integers $D$, we also show that the expected value from the Birch and Swinnerton-Dyer conjecture of the order of the Tate-Shafarevich group is an integer square in certain cases, and an integer square up to a factor $2^{2a}3^{2b}$ in general.

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