Definition 64.19.1. 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 we set
\[ L(X, \mathcal{F}) := \prod \nolimits _{x\in |X|} \det (1 - \pi _ x^*T^{\deg x} |_{\mathcal{F}_{\bar x}})^{-1} \in \Lambda [[ T ]] \]
where $|X|$ is the set of closed points of $X$, $\deg x = [\kappa (x): k]$ and $\bar x$ is a geometric point lying over $x$. This definition clearly generalizes to the case where $\mathcal{F}$ is replaced by a $K \in D_{ctf}(X, \Lambda )$. We call this the $L$-function of $\mathcal{F}$.
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