L-functions depend on characters (or representations), zeta functions do not (or correspond to a trivial character). For example, $L(s,\chi) = \sum_{n = 1}^{\infty}\chi(n)n^{-s}$ where $\chi$ is a Dirichlet character. Supposing that $\chi_0$ is the trivial character modulo $q$, we get $L(s,\chi_0) = \zeta(s)\prod_{p|q}(1 - p^{-s})$ where $\zeta(s)$ is the Riemann zeta function.
So the Dirichlet L-functions generalize the Riemann zeta function. The Dedekind zeta function also generalizes in the same way, to L-functions with Hecke Grössencharacters.
The Dedekind zeta function of
(Removed a number field is not always false statement at the end; as David Hansen points out, one can get a product of factorization into Artin L-functions . After all, a number field can have trivial (belonging to the Galois group.closure of the extension) even when the extension is not Galois by factorizing the Dedekind zeta function of the Galois closure and taking away some factors)
