The S = 3/2, quasi-one-dimensional (1D) zig-zag chain Heisenberg antiferromagnet Li3RuO4 has been investigated using heat capacity, inelastic neutron scattering, neutron diffraction, and μSR measurements on a powder sample. Our neutron diffraction and μSR studies confirm a long-range ordering of the magnetic moments on the Ru5+ cations below 40 K. The magnetic excitations were measured at various temperatures above and below the three-dimensional (3D) ordering temperature in order to understand the broad peak observed in the temperature dependence of the magnetic susceptibility. At 5 K we have observed two well-defined magnetic excitations at 5.5 meV and 8.5 meV and a weak low-energy peak near ~2 meV. We interpret the 5.5 meV energy peak as a 1D zone-boundary mode and that at 8.5 meV as arising from a maximum away from the zone boundary in the dispersion curve for spin-wave modes along the chain of Ru5+ ions. The weaker peak near 2 meV is thought to arise from a weak interchain coupling. Our data are best reproduced using a model with three intrachain interactions and one weak interchain interaction. The experimental spin-exchange interactions are in good agreement with those calculated for a 1D model by density functional theory (DFT) methods. Furthermore, above TN we observe strong diffuse scattering at the same Q-position as the 5.5 meV mode, which suggests the presence of short-range magnetic correlations above TN. We have estimated the correlation length ξ ~2.9 Å at 50 K, which is close to 2.99 Å, the shortest distance between the Ru5+ cations along the zig-zag chain.