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We investigate the impact of the neutron-star matter equation of state on the f- and p1-mode oscillations of neutron stars obtained within the Cowling approximation and linearized general relativity. The f- and p₁-mode oscillation frequencies, and their damping times are calculated using representative sets of Skyrme Hartree-Fock and relativistic mean-field models, all of which reproduce nuclear systematics and support 2M_⊙ neutron stars. Our study shows strong correlations between the frequencies of f- and p₁-modes and their damping times with the pressure of β-equilibrated matter at densities equal to or slightly higher than the nuclear saturation density ρ₀. Such correlations aremore » found to be almost independent of the composition of the stars. The frequency of the p₁-mode of 1.4M_⊙ star is strongly correlated with the slope of the symmetry energy L₀ and β-equilibrated pressure at density ρ₀. Compared to GR calculations, the error in the Cowling approximation for the f-mode is about 30% for neutron stars of low mass, whereas it decreases with increasing mass. Here, the accuracy of the p₁-mode is better than 15% for neutron stars of maximum mass, and improves for lower masses and higher number of radial nodes. I.« less