We derive explicit bounds for the computation of normalizing constants Z for log-concave densities $\pi= \mathrm{e}^{−U} /Z$ w.r.t. the Lebesgue measure on $\mathbb{R}^d$. Our approach relies on a Gaussian annealing combined with recent and precise bounds on the Unadjusted Langevin Algorithm (Durmus, A. and Moulines, E. (2016). High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm). Polynomial bounds in the dimension $d$ are obtained with an exponent that depends on the assumptions made on $U$. The algorithm also provides a theoretically grounded choice of the annealing sequence of variances. A numerical experiment supports our findings. Results of independent interest on the mean squared error of the empirical average of locally Lipschitz functions are established.