Using scanning tunneling spectroscopy we examined the local density of states of thin c-axis La2-xSrxCuO4 films, over wide doping and temperature ranges. We found that the pseudogap exists only at doping levels lower than optimal. For x = 0.12, close to the anomalous x = 1/8 doping level, a zero bias conductance peak was the dominant spectral feature, instead of the excepted V- shaped (c-axis tunneling) gap structure. We have established that this surprising effect cannot be explained by tunneling into (110) facets. Possible origins for this unique behavior are discussed.