The cubic Pell equation $L$-function

For $d \gt 1$ a cubefree rational integer, we define an $L$-function (denoted $L_d(s)$) whose coefficients are derived from the cubic theta function for $\mathbb Q(\sqrt {-3})$. The Dirichlet series defining $L_d(s)$ converges for ${\rm Re}(s) \gt 1$, and