  Let $k=\mathbb{Q}(\sqrt{-3})$, and let $c\in \mathfrak{O}_k$ be a square free
algebraic integer such that $c\equiv 1~({\rm mod}~{\langle9\rangle})$. Let
$\zeta_{k(c^{1/3})}(s)$ be the Dedekind zeta function of the cubic field
$k(c^{1/3})$ and $\zeta_k(s)$ be the Dedekind zeta function of $k$. For fixed
real $\sigma>1/2$, we obtain asymptotic distribution functions $F_{\sigma}$ for
the values of the logarithm and the logarithmic derivative of the Artin
$L$-functions \begin{equation*} L_c(\sigma)=
\frac{\zeta_{k(c^{1/3})}(\sigma)}{\zeta_k(\sigma)}, \end{equation*} as $c$
varies. Moreover, we express the characteristic function of $F_{\sigma}$
explicitly as a product indexed by the prime ideals of $\mathfrak{O}_k$. As a
corollary of our results, we establish the existence of an asymptotic
distribution function for the error term of the Brauer-Siegel asymptotic
formula for the family of number fields $\{k(c^{1/3})\}_{c}$. We also deduce a
similar result for the Euler-Kronecker constants of this family.
