The electronic structure, band gap, density of states of the (8,8), (14,0) and (12,3) single-walled carbon nanotubes by the SIESTA (Spanish Initiative for Electronic Simulations with Thousands of Atoms) method in the framework density-functional theory (DFT) with the generalized gradients approximation (GGA) were studied. Also, we studied the vibrational properties of the (8,8) and (14,0) nanotubes. Only the calculated relaxed geometry for (12,3) nanotube show significant deviations from the ideal rolled graphene sheet configuration. The electronic transition energies of van Hove singularities were studied and compared with previous results. The calculated band structures, density of states and dispersion curves for all tubes were in good agreement with theoretical and experimental results.