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Expansions through the 24th order at high-temperature and up to 11th order atlow-temperature are derived for the main observables of the Blume-Capel modelon bipartite lattices (sq, sc and bcc) in 2d and 3d with various values of thespin and in presence of a magnetic field. All expansion coefficients arecomputed exactly as functions of the crystal and magnetic fields. Severalcritical properties of the model are analyzed in the two most studied cases ofspin S=1 and S=3/2.

The Blume-Capel model for spins S=1 and 3/2 in dimensions d=2 and 3