  We give a conjectural description of the vanishing order and leading Taylor
coefficient of the Zeta function of a proper, regular arithmetic scheme
$\mathcal{X}$ at any integer $n$ in terms of Weil-\'etale cohomology complexes.
This extends work of Lichtenbaum \cite{Lichtenbaum05} and Geisser
\cite{Geisser04b} for $\mathcal{X}$ of characteristic $p$, of Lichtenbaum
\cite{li04} for $\mathcal{X}=\mathrm{Spec}(\mathcal{O}_F)$ and $n=0$ where $F$
is a number field, and of the second author for arbitrary $\mathcal{X}$ and
$n=0$ \cite{Morin14}. We show that our conjecture is compatible with the
Tamagawa number conjecture of Bloch, Kato, Fontaine and Perrin-Riou
\cite{fpr91} if $\mathcal{X}$ is smooth over $\mathrm{Spec}(\mathcal{O}_F)$,
and hence that it holds in cases where the Tamagawa number conjecture is known.
