  For $d > 1$ a cubefree rational integer, we define an $L$-function (denoted
$L_d(s)$) whose coefficients are derived from the cubic theta function for
$\mathbb Q\left(\sqrt{-3}\right)$. The Dirichlet series defining $L_d(s)$
converges for $\text{Re}(s) > 1$, and its coefficients vanish except at values
corresponding to integral solutions of $mx^3 - dny^3 = 1$ in $\mathbb
Q\left(\sqrt{-3}\right)$, where $m$ and $n$ are squarefree. By generalizing the
methods used to prove the Takhtajan-Vinogradov trace formula, we obtain the
meromorphic continuation of $L_d(s)$ to $\text{Re}(s) > \frac{1}{2}$ and prove
that away from its poles, it satisfies the bound $L_d(s) \ll |s|^{\frac{7}{2}}$
and has a possible simple pole at $s = \frac{2}{3}$, possible poles at the
zeros of a certain Appell hypergeometric function, with no other poles. We
conjecture that the latter case does not occur, so that $L_d(s)$ has no other
poles with $\text{Re}(s) > \frac{1}{2}$ besides the possible simple pole at $s
= \frac{2}{3}$.
