For every modulus q ≥ 3, we define a family of subsets A of the multiplicative group (Z/qZ) × for which the Euler product p+qZ∈A (1 − p −s) can be computed with high numerical precision, where s > 1 is some given real number. We provide a Sage script to do so, and extend our result to compute Euler products p+qZ∈A F (1/p s)/H(1/p s) where F and H are polynomials with real coefficients, when this product converges absolutely. This enables us to give precise values of several Euler products occurring in number theory.