  With a super-high-efficient numerical algorithm, we are able to
self-consistently calculate the Green's function in the
renormalized-ring-diagram approximation for a two-dimensional electron system
with long-range Coulomb interactions. The obtained ground-state energy is found
to be in excellent agreement with that of the Monte Carlo simulation. The
numerical results of the self-energy, the effective mass, the distribution
function, and the renormalization factor of the Green's function for the
coupling constants in the range $0 \le r_s \le 30$ are also presented.
