In this paper, we consider the eigenvalue problem for the equation of a vibrating beam with a spectral parameter contained in the boundary conditions. The general characteristics of the location of eigenvalues on the real axis are studied, the multiplicities of eigenvalues are found, and the oscillatory properties of the eigenfunctions of this spectral problem are investigated. Moreover, asymptotic formulas for the eigenvalues are obtained and sufficient conditions are established for the subsystems of root functions to form a basis in Lebesgue spaces.