  We present a detailed investigation of the specific heat in
Ca$_3$(Ru$_{1-x}M_x$)$_2$O$_7$ ($M$ = Ti, Fe, Mn) single crystals. With
different dopants and doping levels, three distinct regions are present,
including a quasi-2D metallic state with an antiferromagnetic (AFM) order
formed by ferromagnetic bilayers (AFM-$b$), a Mott insulating state with G-type
AFM order (G-AFM) and a localized state with a mixed AFM-b and G-AFM phase. Our
specific heat data provide deep insights into the Mott transitions induced by
Ti and Mn dopings. We observed not only an anomalous large mass enhancement but
also an additional term in the specific heat i.e. $C\propto T^2$ in the
localized region. The $C\propto T^2$ term is most likely due to the
long-wavelength excitations with both FM and AFM components. A decrease of
Debye temperature is observed in the G-type AFM region, indicating a lattice
softening associated with the Mott transition.
