Abstract: 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 refinement which equates invariants only depending on the Brauer group to invariants only depending on the Neron-Severi group. We also give estimates of the terms appearing in the formula. This implies, for example, the largest Brauer group of an abelian surface over the field of order q=p^{2r} has order 16q, and the largest Brauer group of a supersingular abelian surface over a prime field is 36.