Computing a dispersion relation for warm plasma waves generally requires numerical methods since the relation is expressed as an implicit function of frequency and wavenumber. In this work, we present an explicit dispersion relation that provides an accurate analytical solution for the dispersion without relying on numerical methods. The derived formula can even describe the dispersion relation near the cyclotron resonance that the implicit relation cannot represent. We verify the validity and accuracy of the formula by comparing its analytical solutions with the numerical results obtained from the implicit dispersion relations and fully kinetic particle-in-cell (PIC) simulations.