  We investigate the distribution of values of cubic Dirichlet $L$-functions at
$s=1$. Following ideas of Granville and Soundararajan for quadratic
$L$-functions, we model the distribution of $L(1,\chi)$ by the distribution of
random Euler products $L(1,\mathbb{X})$ for certain family of random variables
$\mathbb{X}(p)$ attached to each prime. We obtain a description of the
proportion of $|L(1,\chi)|$ that are larger or that are smaller than a given
bound, and yield more light into the Littlewood bounds. Unlike the quadratic
case, there is an asymmetry between lower and upper bounds for the cubic case,
and small values are less probable than large values.
