  Arising from the factorizations of Dedekind zeta-functions of cubic fields,
we obtain Artin $L$-functions of certain two-dimensional representations. In
this paper, we study the value-distribution of such Artin $L$-functions for
families of non-Galois cubic fields in conductor aspect. We prove that various
mean values of the Artin $L$-functions are represented by integrals involving a
density function which can be explicitly constructed. By the class number
formula, the result is applied to the study on the distribution of class
numbers of cubic fields.
