The influence of ohmic (metallic) loss on the complex band structure (BS) and eigenmodes of 2-D plasmonic crystals is investigated. With the help of wave equations and periodic boundary conditions, a finite-difference-based eigenvalue algorithm is proposed to model the plasmonic crystals with arbitrarily lossy and dispersive materials. Given a frequency of interests, the algorithm solves one complex Bloch wavenumber as the eigenvalue via fixing another. Most importantly, the developed eigenvalue analysis could expand the bulk excitation solution with eigenmodes, which satisfies the generalized phase (momentum) matching condition. For a TE polarization with $H_{z}$ field, the ohmic loss strongly affects the BS and eigenmodes at plasmonic resonance frequencies. Both the fast oscillation of a dispersion curve and strong field confinement of eigenmodes are damped due to the high ohmic loss. For a TM polarization with $E_{z}$ field, the introduction of ohmic loss twists the vertical dispersion curve at the bandgap and breaks the symmetry of the eigenmodes. For both polarizations, the high ohmic loss lowers the quality factor of the eigenmodes. This paper offers a fundamental and important eigenvalue analysis for designing lossy and dispersive plasmonic crystals.