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


We have $\zeta\_F(s) = \frac{\lambda\_{-1}}{s-1} + \lambda\_0 + \dots$ 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 $1$.



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> What is known or conjectured about $\lambda\_0$?
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Tate's thesis can be copied word by word for function fields with the same class number formula, so:



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> What is known for the zeta function of a function field?
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Dedekind Zeta function: behaviour at 1
This is called the (generalised) Euler constant of the number field $K$, denoted $\gamma\_K$, as for $K = \mathbb{Q}$ we have $\gamma\_K = \gamma\_0$, the Euler--Mascheroni constant. There are many estimates known for $\gamma\_K$. For example, Theorem 7 of [this paper](https://dx.doi.org/10.1016/j.exmath.2006.08.001) has an upper bound for $|\gamma\_K|$. Some other useful references are Theorem B of [this paper](https://projecteuclid.org/journals/bulletin-of-the-belgian-mathematical-society-simon-stevin/volume-11/issue-4/Eulers-constants-for-the-Selberg-and-the-Dedekind-zeta-functions/10.36045/bbms/1102689119.full) as well as [this paper](https://dx.doi.org/10.1017/S0013091500015844).


As for a function field, I am not so sure, but I'm guessing things should be similar but slightly easier (as there are only finitely many zeroes for $\zeta\_{C/\mathbb{F}\_q}(s)$).


EDIT: [This paper](https://dx.doi.org/10.1007/978-0-8176-4532-8_5) deals with bounds for $\gamma\_K$ for function fields.


