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Dispersion and attenuation characteristics of guided and leaky modes on circular lossy dielectric rod waveguides are analyzed for several lower order modes using an efficient and accurate Davidenko's complex root finding algorithm. In the leaky mode region, the normalized phase and attenuation constants of the lossless and lossy cases are very similar compared with the lossless case, if the dielectric is low loss, thereby the performance of the leaky wave antenna and other physical phenomena in the structure are expected to be little affected. The normalized phase constant of the guided mode has deflected to lower values compared with the lossless case as the operating frequency goes down, and finally the guided mode becomes cutoff or evolves into the next higher order leaky mode in the spectral gap region of that leaky mode. It is also observed that the attenuation constants obtained from the traditional perturbation method are meaningful only in some limited frequency ranges.