explicit lower bounds on $|L(1,\chi)|$

Does anyone know of an explicit effective lower bound for $|L(1,\chi)|$, where $\chi$ is an odd complex (primitive) Dirichlet character?

I know of Landau's paper Uber Dirichletsche Reihen mit komplexen Charakteren, where he bounds $$|L(1,\chi)|>\frac{1}{c \log(q)},$$

where $q$ is the conductor of $\chi$, but the constant $c$ he gets is on the order of $e^{50}$, and is totally useless for computations.

I know of many papers dealing with quadratic characters but very few that address complex characters (explicitly).

This is discussed on page 47 of Narkiewicz's new book (Rational Number Theory in the 20th Century); see