Brauer groups and Neron-Severi groups of surfaces over finite fields

For a smooth and proper surface over a finite field, the formula of Artin and Tate relates the behavior of the zeta-function at 1 to other invariants of the surface. We give a version which equates invariants only depending on the Brauer group to invariants only depending on the Neron-Severi group. We estimate the terms appearing in the formula, and discuss the special case of abelian varieties and K3-surfaces.