Kaneko–Koyama–Kurokawa Deep Riemann Hypothesis

Let $\pi$ be an irreducible cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_\mathbb Q)$, with entire $L$-function

$$L(s,\pi)=\prod_p\prod_{j=1}^n(1-\alpha_{j,p}p^{-s})^{-1}.$$

Let $\nu(\pi)=m(\operatorname{sym}^2\pi)-m(\wedge^2\pi)\in\mathbb Z$, where $m(\rho)$ is the multiplicity of the trivial representation in $\rho$, and let $m=\operatorname{ord}_{s=1/2}L(s,\pi)$. Kaneko–Koyama–Kurokawa conjecture. The limit

$$\lim_{x\to\infty}(\log x)^m\prod_{p\leq x}\prod_{j=1}^n\left(1-\alpha_{j,p}p^{-1/2}\right)^{-1}$$

exists and is nonzero, and equals

$$\frac{\sqrt{2}^{\,\nu(\pi)}}{e^{m\gamma}m!}L^{(m)}\left(\frac12,\pi\right).$$

The source calls this the Deep Riemann Hypothesis because it is intended to lie deeper than, and imply, the Generalized Riemann Hypothesis for these automorphic $L$-functions. Its resolution status is not specified in the supplied text.