We will show how to combine the Fast Fourier Transform algorithm with the reflection formulae of the special functions involved in the computation of the values of $L(1,chi)$ and $L'(1,chi)$, where $chi$ runs over the Dirichlet characters modulo an odd prime number $q$. In this way, we will be able to reduce the memory requirements and to improve the computational cost of the whole procedure. Several applications to number-theoretic problems will be mentioned, like the study of the distribution of the Euler-Kronecker constants for the cyclotomic field and its subfields, the behaviour of $min_{chie chi_0} | L'(1,chi)/L(1,chi) |$, the study of the Kummer ratio for the first factor of the class number of the cyclotomic field and the ``Landau vs. Ramanujan`` problem for divisor sums and coefficients of cusp forms. Towards the end of the seminar we will tackle open problems both of theoretical and implementative nature.