  We investigate the thermodynamic properties of the zero-field Blume-Capel
model in the vicinity of its tricritical point (TCP). We calculate the
quadrupole moment, internal energy, and entropy densities employing an exact
numerical recursion procedure for the model defined on a hierarchical lattice
of fractal dimension $d$. We explore the scaling behavior of the isothermal
quadrupolar susceptibility and, the isothermal and constant crystal-field
specific heat as a function of the temperature and the reduced crystal-field
parameter along the ferromagnetic and the \emph{ordered paramagnetic} phase
frontier. Results achieved for systems with dimensions $d=2$ and $3$ exhibit
the main features of the continuous and first-order transitions in the TCP
neighborhoods. We also probe the phase coexistence in the $\lambda$-diagram and
the latent heat in the vicinity of the tricritical point locus.
