The turbulent Riemann hypothesis for generalized L-functions

Let $f$ and $g$ be functions from $\mathbb{N}\times\mathbb{N}$ to $\mathbb{R}$, with the coefficient and exponent data described by the source. Define the generalized $L$-function

$$\zeta_{(f,g)}(z)=\sum_{i,j}f_{ij}\frac{\partial^{g_{ij}j^{-z}}}{\partial z^{g_{ij}j^{-z}}}i^{-z}.$$

Let $\lambda(g)$ be the average of the eigenvalues of the $g_{ij}$, weighted by multiplicity.

Turbulent Riemann hypothesis. The analytic extension of $\zeta_{(f,g)}$ to the complex numbers has nontrivial zeroes only on the critical line

$$\operatorname{Re}(z)=\left.\frac{d}{dx}\arctan\left(\sqrt{x\tan(x)}\right)\right|_{x\tan(x)=\lambda(g)}.$$

This is explicitly presented as a conjectural consequence of the preceding variational analysis. The generalized function and its fractional derivative notation are not rigorously developed in the supplied text.