In this paper, we study the initial value problem of longitudinal plasma oscillations in a relativistic electron beam. Our analysis is based on the formalism developed in Marinelli et al. [Phys. Plasmas 18, 103105 (2011)]. We study the evolution of an arbitrary six-dimensional phase-space perturbation under the effect of longitudinal space-charge forces, with the inclusion of three-dimensional effects due to the finite size of the beam, transverse betatron motion, and longitudinal thermal motion induced by both energy-spread and transverse emittance. We expand the phase-space perturbation in a series of eigenmodes of a Schrödinger-like equation, corresponding to a set of propagating space-charge waves. We develop a general formalism, which we use to find explicit expressions for the evolution of an initial perturbation coupled to the fundamental plasma eigenmode. This work has important applications in the theory of space-charge instabilities in high brightness electron beams and control of shot-noise in seeded free-electron lasers. We discuss the application of the present theory to a specific experimental scenario corresponding to a shot-noise suppression scheme at optical wavelengths.