The Riemann zeta function is \zeta(s) = \sum_{n\geq 1} n^{-s} = \prod_{\ell}\left(1 - \ell^{-s}\right)^{-1}, the product — an Euler product — running over all primes \ell; the second equality expresses unique factorisation of integers. The sum converges absolutely on the right half-plane \set{s\in\C : \mathrm{Re}(s) > 1}, where it defines a holomorphic function. It admits a meromorphic continuation to all of \C and satisfies a functional equation relating \zeta(s) and \zeta(1-s).

Let F be a number field with ring of integers \roi_F. The Dedekind zeta function of F is \zeta_F(s) = \sum_{0\neq I\subseteq\roi_F} \Nm(I)^{-s} = \prod_{\mathfrak p}\left(1 - \Nm(\mathfrak p)^{-s}\right)^{-1}, where I runs over the non-zero ideals of \roi_F and \mathfrak p over the non-zero prime ideals. It converges for \mathrm{Re}(s)>1, continues meromorphically to \C, and satisfies a functional equation relating \zeta_F(s) and \zeta_F(1-s). The Euler product again reflects unique factorisation of ideals. (Mathlib's NumberField.dedekindZeta is exactly this Dirichlet series; the meromorphic continuation is not yet formalised there, though the residue behaviour at s=1 is.)

Let \chi:(\Z/N\Z)^{\times}\to\C^{\times} be a Dirichlet character. Extend \chi to \chi:\Z\to\C by \chi(m)=\chi(m\bmod N) when (m,N)=1 and \chi(m)=0 otherwise. The Dirichlet L-function of \chi is L(\chi,s) = \sum_{n\geq 1}\chi(n)n^{-s} = \prod_{\ell}\left(1 - \chi(\ell)\ell^{-s}\right)^{-1}. It converges for \mathrm{Re}(s)>1, continues meromorphically to \C (analytically when \chi is non-trivial), and satisfies a functional equation relating s and 1-s. The Riemann zeta function is the case \chi=1.

Let E/\Q be an elliptic curve of conductor N, and write a_\ell(E)=\ell+1-\#E(\F_\ell) for the trace of Frobenius at a good prime \ell (with \F_\ell the field of \ell elements). The L-function of E is L(E,s) = \sum_{n\geq 1}a_n(E)n^{-s} = \prod_{\ell\nmid N}\left(1 - a_\ell(E)\ell^{-s} + \ell^{1-2s}\right)^{-1}\prod_{\ell\mid N}L_\ell(s), the coefficients a_n(E) being built recursively from the a_\ell(E), and the bad Euler factors L_\ell(s) depending on the reduction type at \ell. The sum converges for \mathrm{Re}(s)>3/2, continues analytically to \C, and satisfies a functional equation relating s and 2-s.

Let f=\sum_{n\geq 1}a_n(f)q^n\in S_k(\Gamma_0(N),\Teich_f) be a normalised newform of weight k, level N and nebentypus \Teich_f. The L-function of f is L(f,s) = \sum_{n\geq 1}a_n(f)n^{-s} = \prod_{\ell\nmid N}\left(1 - a_\ell(f)\ell^{-s} + \Teich_f(\ell)\ell^{k-1-2s}\right)^{-1}\prod_{\ell\mid N}\left(1 - a_\ell(f)\ell^{-s}\right)^{-1}. It converges for \mathrm{Re}(s)>(k+1)/2, continues analytically to \C, and satisfies a functional equation relating s and k-s.

More generally, let \cG_\Q=\Gal(\Qbar/\Q) and let V be a p-adic Galois representation, i.e. a finite-dimensional vector space over a finite extension L of \Qp with a continuous linear \cG_\Q-action. Its global L-function is the formal Euler product L(V,s)=\prod_\ell L_\ell(V,s) of local factors. For \ell\neq p one sets L_\ell(V,s) = \det\!\left(\mathrm{Id} - \Frob_\ell^{-1}\ell^{-s}\,\middle|\, V^{I_\ell}\right)^{-1}, with \Frob_\ell the arithmetic Frobenius and I_\ell the inertia at \ell; at \ell=p one uses the crystalline module, L_p(V,s)=\det(\mathrm{Id}-\varphi^{-1}p^{-s}\mid\Dcris(V))^{-1}. When V is the representation attached to an arithmetic object the two notions of L-function agree; for V=\Qp(\chi) one recovers the Dirichlet L-function Definition 2.1.3 of \chi.