Goldfeld–Hinkle pole-free continuation conjecture for the cubic Pell equation L-function

Fix $\varepsilon > 0$ and let $d>1$ be a cubefree rational integer. Let $L_d(s)$ denote the cubic Pell equation $L$-function.

Goldfeld–Hinkle's pole-free continuation conjecture. The function $L_d(s)$ has meromorphic continuation to $\textup{Re}(s)>\frac12$ with at most a simple pole at $s=\frac23$. In the region $\textup{Re}(s)>\frac12+\varepsilon$ and $\left|s-\frac23\right|>\varepsilon$, it satisfies

$$L_d(s) \ll_{d,\varepsilon} |s|^{\frac72}.$$

This is the conjectural consequence of the asserted nonvanishing of the Appell hypergeometric factor in the preceding theorem. It is a restatement of the preceding pole conjecture together with the already established growth estimate, rather than a separate mathematical claim.