Cube finite-place mean conjecture

Let $C$ be the determinant-admissible set associated with the cubic polynomial $P_3(n)$, and let $N!_C$ denote its factorial. Define the local slope, when it exists, by $c_p$, with

$$c_3=\sqrt6-\frac32,$$

and, for $p\equiv2\pmod3$,

$$c_p=\frac{-3+\sqrt{9+\dfrac{12p}{(p-1)^2}}}{2}.$$

Let $\chi_{-3}$ be the quadratic character of conductor $3$.

Cube finite-place mean conjecture. The limit

$$\beta_C=\lim_{N\to\infty}\frac1N\log\frac{N!_C}{\sqrt{N!(3N)!}}$$

exists and equals

$$\begin{aligned} \beta_C&=\left(\sqrt6-\frac52\right)\log3-\frac{L'}{L}(1,\chi_{-3})\\ &\quad+\sum_{p\equiv2\pmod3}\left(c_p-\frac1{p-1}-\frac2{p^2-1}\right)\log p. \end{aligned}$$

The remaining prime sum is absolutely convergent.

This conjecture gives the asymptotic mean of the factorial attached to the cubic determinant-admissible set, including an explicit contribution from the quadratic character of conductor $3$ and the inert-prime renewal slopes. The stated context presents the constant as a prediction; the existence of the limit, the displayed evaluation, and absolute convergence of the residual prime sum remain to be proved.