Low-temperature specific heat of (La1-xSrx)2CuO4 has been studied between 2 and 20 K by systematically varying the Sr concentration x, in order to clarify the origin of non-zero γ, the temperature-linear specific-heat coefficient, frequently observed in high-Tc superconducting oxides. Identical procedure is used to prepare samples for the specific-heat measurement and for the determination of the structural, magnetic and superconductive phase-transition temperatures as a function of x. The value of γ is nearly zero in the antiferromagnetic ordering region, but increase abruptly with x to about 4 mJ/K2 mol in the insulating, but not antiferromagnetic region. By increasing x further into the superconductive region, γ tends to be smaller: 1.6 mJ/K2mol for x = 0.075. We argue that the non-zero γ is not due to inhomogeneity of the samples, but due to electronic origin intrinsic to this system.