A 0.25 g single crystal of Bi2Sr2CaCu2O8 with a sharp superconducting transition at 85 K was characterized by AC susceptibility, X-ray diffraction, Meissner effect, normal-state susceptibility, magnetization in the mixed state, and specific heat from 1 to 300K. Many unconventional features are found in the specific heat which may be intrinsic features of a 2D superconductor, e.g. a field-independent linear term γ∗T at low T, the absence of a specific heat jump at Tc, a symmetrical logarithmic anomaly at Tc, an almost field-independent transition temperature which coincides with the crossing point of the magnetization, a new crossing point of the specific heat, etc. The accurate data near Tc (<0.02%) allow an expression of ∂(C/T)/∂H by an empirical formula that describes the variation of the specific heat and magnetization over a wide temperature and field range about Tc.