Let $\zeta$ denote the Dedekind zeta function of a number field $F$.

We have $\zeta(s) = \frac{\lambda_{-1}}{s-1} + \lambda_0 + \dots$ for $s-1$ small.

Class number formula: We have $\lambda_{-1} = vol( F^\times \backslash \mathbb{A}^1)$, where $\mathbb{A}^1$ denotes the group of ideles with norm $1$.

What is known or conjectured about $\lambda_0$?

Tate's thesis can be copied word by word for function fields with the same class number formula, so:

What is known for the zeta function of a function field?