Theorem 63.20.1 (Finite Coefficients). Let $X$ be a scheme of finite type over a finite field $k$. Let $\Lambda$ be a finite ring of order prime to the characteristic of $k$ and $\mathcal{F}$ a constructible flat $\Lambda$-module on $X_{\acute{e}tale}$. Then

$L(X, \mathcal{F}) = \det (1 - \pi _ X^*\ T |_{R\Gamma _ c(X_{\bar k}, \mathcal{F})})^{-1} \in \Lambda [[T]].$

Proof. Omitted. $\square$

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