  Let $K/\mathbb Q$ be a finite Galois extension and let $\chi_1,\ldots,\chi_r$
be the irreducible characters of the Galois group $G:=Gal(K/\mathbb Q)$. Let
$f_1:=L(s,\chi_1),\ldots,f_r:=L(s,\chi_r)$ be their associated Artin
L-functions. For $s_0\in \mathbb C\setminus\{1\}$, we denote $Hol(s_0)$ the
semigroup of Artin $L$-functions, holomorphic at $s_0$. Let $\mathbb F$ be a
field with $\mathbb C \subseteq \mathbb F \subseteq \mathcal M_{<1}:=$ the
field of meromorphic functions of order $<1$. We note that the semigroup ring
$\mathbb F[Hol(s_0)]$ is isomorphic to a toric ring $\mathbb F[H(s_0)]\subseteq
\mathbb F[x_1,\ldots,x_r]$, where $H(s_0)$ is an affine subsemigroup of
$\mathbb N^r$ minimally generated by at least $r$ elements, and we describe
$\mathbb F[H(s_0)]$ when the toric ideal $I_{H(s_0)}=(0)$. Also, we describe
$\mathbb F[H(s_0)]$ and $I_{H(s_0)}$ when $f_1,\ldots,f_r$ have only simple
zeros and simple poles at $s_0$.
