We investigate the impact of the neutron-star matter equation of state on the - and -mode oscillations of neutron stars obtained within the Cowling approximation and linearized general relativity. The - and -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 neutron stars. Our study shows strong correlations between the frequencies of - and -modes and their damping times with the pressure of -equilibrated matter at densities equal to or slightly higher than the nuclear saturation density . Such correlations are found to be almost independent of the composition of the stars. The frequency of the -mode of star is strongly correlated with the slope of the symmetry energy and -equilibrated pressure at density . Compared to GR calculations, the error in the Cowling approximation for the -mode is about 30% for neutron stars of low mass, whereas it decreases with increasing mass. The accuracy of the -mode is better than 15% for neutron stars of maximum mass, and improves for lower masses and higher number of radial nodes.