The instantaneous-normal-mode (INM) densities of states for liquid Na in four realistic thermodynamic states are obtained by molecular-dynamics simulation. The exponent functions derived from the INM densities of states are found to fit well with a binomial of frequency for the imaginary-frequency lobe and a three-term polynomial for the real-frequency lobe. Through the study of the size dependence for the participation ratios of INMs, we show that for both the real-frequency and the imaginary-frequency lobes, INMs of low frequencies are extended modes and INMs in the high-frequency end are localized modes. However, each lobe has a broad transition region in frequency from the extended to localized INMs.