In this paper, we give a direct quantitative estimate of $L^1$ norms of non-harmonic trigonometric polynomials over large enough intervals. This extends the result by Konyagin and Mc Gehee, Pigno, Smith to the setting of trigonometric polynomials with non-integer frequencies. The result is a quantitative extension of a result by Nazarov and also covers a result by Hudson and Leckband when the length of the interval goes to infinity.