We provide a number of results that can be used to derive approximations for the Euler product representation of the zeta function of an arbitrary algebraic function field. Three such approximations are given here. Our results have two main applications. They lead to a computationally suitable algorithm for computing the class number of an arbitrary function field. The ideas underlying the class number algorithms in turn can be used to analyze the distribution of the zeros of its zeta function. We will present numerical data that provides shows evidence for the effectiveness of the class number algorithm in the case of purely cubic function fields over large prime fields. We also provide interesting various data for the distribution of the zeros of the zeta function of those cubic function fields. This is based on recent joint work with Bauer, Landquist, and Scheidler.