Lichtenbaum–Milne conjecture on special values of zeta functions

Let $X$ be a smooth projective geometrically connected variety over $\mathbb{F}_q$, let $Z(X,t)$ be defined by $\zeta(X,s)=Z(X,q^{-s})$, and let $\rho_r$ be the order of the pole of $\zeta(X,s)$ at $s=r$. Let $\chi(X,\mathbb{Z}(r))$ and $\chi(X,\mathcal{O},r)$ denote the invariants defined by Lichtenbaum and Milne through motivic cohomology and Hodge-cohomological data, respectively.

Lichtenbaum–Milne conjecture. For every relevant integer $r$, as $t\to q^{-r}$,

$$Z(X,t)\sim \pm\chi(X,\mathbb{Z}(r))q^{\chi(X,\mathcal{O},r)}(1-q^rt)^{-\rho_r}.$$

Equivalently, the special value $a_X(r)$ is described by an Euler characteristic of motivic cohomology, including its $p$-part. The precise definitions of the terms are referred to the cited work of Milne; the conjecture is not known in general.