Does there exist a two variable analogue of the Weil conjecture?

What I mean is that the usual Weil involves a one-variable zeta-function which you get by using numbers $V_n = V ( GF(p^n))$ of points of a smooth algebraic variety over finite fields of characteristic $p$. Is it possible to have a sensible two-parameter family of finite rings instead? Any references?

For instance, one can consider finite quotients of Witt vectors, and form a two-parameter family of numbers $V_{n,m} = V ( Witt(GF(p^n))/I^m)$ (where $I$ is the maximal ideal of the Witt vectors) from a variety $V$ (smooth, projective over $\mathbb Z$). Is there a sensible two variable zeta-function cooked with these numbers?