$L$-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 $L(E_{D, \alpha}, 1)$correspondingtothefamilyofsextictwistsover$$\mathbb{Q}$[\sqrt{-3}]$ofellipticcurves$E_{D, \alpha}: y^2=x^3+16D^2\alpha^3$for$D$anintegerand$\alpha \in $\mathbb{Q}$[\sqrt{-3}]$\mathrm{.}Theformulageneralizestheresultofcubictwistsover$$\mathbb{Q}$$ofRodriguez-VillegasandZagierforaprime$D \equiv 1 (9)$ and of Rosu for general $D$. For $\alpha$ 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.