I've been learning more about different $p$-adic geometries, namely Berkovich spaces, Huber's Adic spaces and ridgid analytic spaces. In arithmetic geometry, it is often very interesting to assoicate and study the L-function of an space. My question is if this has been studied for such $p$-adic spaces. One way one could naively hope to get such L-functions is to take the L-function associated to the cohomology of such a space and study that. However, I've been unable to find a reference for this.
The issue is that google searches lead straight to $p$-adic L-functions, which as far I can tell is not what I want.