Effective Estimates for the Distribution of Values of Euler Products

We prove effective upper bounds for the almost periodicity of polynomial Euler products L(s){cal L}(s) in the half-plane of absolute convergence. From this we deduce estimates for the roots of the equation L(s) = c{cal L}(s) = c, where c is any non-zero complex number which is attained by L(s){cal L}(s). The method relies mainly on effective diophantine approximation.