It is well know that free-space laser system performance is limited by atmospheric turbulence. Most theoretical treatments have been described for many years by Kolmogorov's power spectral density model because of its simplicity. Unfortunately, several experiments have been reported recently that show that the Kolmogorov theory is sometimes incomplete to describe atmospheric statistics properly, in particular, in portions of the troposphere and stratosphere. We present a non-Kolmogorov power spectrum that uses a generalized exponent instead of constant standard exponent value 11/3, and a generalized amplitude factor instead of constant value 0.033. Using this new spectrum in weak turbulence, we carry out, for a horizontal path, an analysis of long-term beam spread, scintillation index, probability of fade, mean signal-to-noise ratio (SNR), and mean bit error rate (BER) as variation of the spectrum exponent. Our theoretical results show that for alpha values lower than α=11/3, but not for alpha close to α=3, there is a remarkable increase of scintillation and consequently a major penalty on the system performance. However, when alpha assumes a value close to α=3 or for alpha values higher than α=11/3, scintillation decreases, leading to an improvement on the system performance.