Grand Riemann hypothesis

Let $F$ be a number field with ring of adeles $\mathbb{A}_F$, and let $n \ge 1$. Let $\pi = \otimes_v \pi_v$ be a cuspidal automorphic representation of $GL_n(\mathbb{A}_F)$ with unitary central character, normalised so that its standard (finite-part) $L$-function

$$L(s,\pi) \;=\; \prod_{v \nmid \infty} L(s,\pi_v),$$

given by the Euler product of the local factors $L(s,\pi_v)$ over the finite places of $F$, converges absolutely for $\operatorname{Re}(s)$ sufficiently large, extends to a meromorphic function on $\mathbb{C}$, and the completed $L$-function

$$\Lambda(s,\pi) \;=\; L(s,\pi_\infty)\,L(s,\pi),$$

formed with the archimedean factor $L(s,\pi_\infty)$, satisfies the functional equation

$$\Lambda(s,\pi) \;=\; \varepsilon(s,\pi)\,\Lambda(1-s,\tilde{\pi}),$$

where $\tilde{\pi}$ is the contragredient of $\pi$ and $\varepsilon(s,\pi)$ is the associated epsilon factor, nowhere vanishing.

Call $s \in \mathbb{C}$ with $0 < \operatorname{Re}(s) < 1$ and $L(s,\pi) = 0$ a non-trivial zero of $L(s,\pi)$.

Then for every number field $F$, every $n \ge 1$, and every such $\pi$,

$$L(s,\pi) = 0 \ \text{ and } \ 0 < \operatorname{Re}(s) < 1 \quad \Longrightarrow \quad \operatorname{Re}(s) = \tfrac{1}{2}.$$

In the modified form, the conclusion is weakened to

$$L(s,\pi) = 0 \ \text{ and } \ 0 < \operatorname{Re}(s) < 1 \quad \Longrightarrow \quad \operatorname{Re}(s) = \tfrac{1}{2} \ \text{ or } \ s \in \mathbb{R}.$$

The case $F = \mathbb{Q}$, $n = 1$ with $\pi$ the trivial character gives $L(s,\pi) = \zeta(s)$, and the cases $F = \mathbb{Q}$, $n = 1$ with $\pi$ a Dirichlet character $\chi$ give $L(s,\pi) = L(s,\chi)$.