[Math] étale cohomology with G_m coefficients

Question

Most calculations of étale cohomology in Milne's book deal with constructible or torsion sheaves. Are there references where the cohomology of varieties with $\mathbf{G}_m$ coefficients are calculated? I'm especially interested in the top dimension $2\mathrm{dim}(X)$ ($+ 1$). I found some calculations in Le groupe de Brauer (In: Dix Exposés sur la Cohomologie des Schémas), but they don't help me.

Edit: Assume $X$ is a variety over a finite field.

Answer 1

I found calculations in S. Lichtenbaum, Zeta functions of varieties over finite fields at s = 1, Arithmetic and geometry, Vol. I, 173–194 Progr. Math., 35, Birkhauser Boston, Boston, MA, 1983, especially Proposition 2.1 and Theorem 2.2--2.4.