For a scheme $X$, people sometimes use $|X|$ to denote the set of closed points of $X$. So the set of primes is $|\operatorname{Spec}(\mathbb{Z})|$ and you have:$$\zeta(s)=\prod_{p\in|\operatorname{Spec}(\mathbb{Z})|}\frac{1}{1-p^{-s}}.$$
This formula of course generalizes to give the $\zeta$-function of any scheme $X$ of finite type over $\operatorname{Spec}(\mathbb{Z})$: \mathbb{Z}$(e.g., a variety of finite type over a finite field): $$\zeta_X(s)=\prod_{x\in|X|}\frac{1}{1-|\kappa(x)|^{-s}}$$ where$\kappa(x)$is the residue field at$x$and$|\kappa(x)|$is its order. 1 [made Community Wiki] For a scheme$X$, people sometimes use$|X|$to denote the set of closed points of$X$. So the set of primes is$|\operatorname{Spec}(\mathbb{Z})|$and you have:$$\zeta(s)=\prod_{p\in|\operatorname{Spec}(\mathbb{Z})|}\frac{1}{1-p^{-s}}.$$ This formula of course generalizes to give the$\zeta$-function of any scheme$X$of finite type over$\operatorname{Spec}(\mathbb{Z})$: $$\zeta_X(s)=\prod_{x\in|X|}\frac{1}{1-|\kappa(x)|^{-s}}$$ where$\kappa(x)$is the residue field at$x$and$|\kappa(x)|\$ is its order.