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We compute the complete set of candidates for the zeta function of a K3surface over F_2 consistent with the Weil conjectures, as well as the completeset of zeta functions of smooth quartic surfaces over F_2. These sets differsubstantially, but we do identify natural subsets which coincide. This givessome numerical evidence towards a Honda-Tate theorem for transcendental zetafunctions of K3 surfaces; such a result would refine a recent theorem ofTaelman, in which one must allow an uncontrolled base field extension.

A census of zeta functions of quartic K3 surfaces over F_2