On-axis intensity fluctuations of laser array beams are evaluated when they are used in a weakly turbulent non-Kolmogorov atmosphere. Our formulation of the scintillation index is based on the Rytov method, which in the limiting case, correctly reduces to the known Gaussian beam scintillation index in weak Kolmogorov turbulence. When the radius of the ring (on which the array beamlets are placed), number of beamlets forming the laser array, source size of the beamlets, propagation distance or the wavelength is fixed, a decrease in the power law exponent of the non-Kolmogorov spectrum is found to decrease the scintillation index. Examining for any realization of the non-Kolmogorov spectrum, it is observed that an increase in the ring radius, number of beamlets, wavelength, and decrease in the source size of the beamlet, propagation distance reduces the intensity fluctuations.