  This note gives a simple proof that certain values of Artin's $L$-function,
for a representation $\rho$ with character $\chi_\rho$, are stable under
twisting by an even Dirichlet character $\chi$, up to an element generated over
$\mathbb Q$ by the values of $\chi$ and $\chi_\rho$, and a product with a power
of the Gauss sum $\tau(\chi)$ equal to the dimension of $\rho$. This extends a
result due to J. Coates and S. Lichtenbaum.
