We study the Euler equations of compressible isentropic gas dynamics with spherical symmetry. Due to the presence of the singularity at the origin, little is known in the case including the origin. In this paper, we prove the existence of local solutions for the case including the origin. A modified Godunov scheme is introduced to construct approximate solutions and obtain $L^{\infty}$ estimates. The convergence and consistency of the approximate solutions are proved with the aid of a compensated compactness framework.