The study of Dirichlet series and their analytic properties lies at the heart of number theory, with deep connections to the distribution of primes and arithmetic invariants of number fields. A fundamental example is the L-series associated with Dirichlet characters, which generalizes the Riemann zeta function and play a crucial role in Dirichlet’s theorem on primes in arithmetic progressions. The behavior of these L-series at s = 1 is particularly significant, as shown in Dirichlet’s class number formula for quadratic fields, which links the value at s = 1 of a L-series to the class number via the Dedekind zeta function of a quadratic number field. Key to evaluating these L-series at s = 1 are Gauss sums, particularly the determination of the sign of the quadratic Gauss sum. We carry out this computation using the discrete Fourier transform following Ram Murty’s approach. This computation also yields another proof of quadratic reciprocity, formulated directly for the Jacobi symbol.