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In this talk, we survey some recent results on the distribution of values of various families of $L$-functions in the critical strip. We also describe a value-distribution theorem for the logarithms and logarithmic derivatives of a family of $L$-functions attached to cubic Hecke characters. As a corollary we establish the existence of an asymptotic distribution function for the error term of the Brauer-Siegel asymptotic formula for a certain family of cubic extensions of $\mathbb{Q}(\sqrt{-3})$. We also deduce a similar result for the Euler-Kronecker constants of this family. This is joint work with Amir Akbary.