Relativistic electrostatic oscillations in a collisionless nonmagnetized plasma are considered. An integral form of the Vlasov–Poisson problem is solved numerically and both the transient and the asymptotic behavior of the electric field are calculated in a very accurate way. The existence of a critical value for the wave number, under which no Landau damping occurs, has been verified, and the asymptotic behavior of the oscillations in the one-dimensional kinetic model has been evaluated. Some errors have been found in previously published analytical expressions. A new expression is proposed and verified numerically. An analytical expression of the asymptotic behavior for full three-dimensional kinetic model is deduced and numerically tested.