We have measured the specific heat of poly- and single-crystalline LaPt3Si samples in various magnetic fields. In zero magnetic field, we observed distinct superconducting transitions at T-c similar to 0.64 K and 0.61 K for the poly- and single crystals, respectively. Temperature T dependences of the specific heat C of both samples around T-c resembled each other and could be well described by an exponential equation for a conventional superconductor at low temperatures. In a magnetic field, a characteristic peak of C/T appeared while the T-c was considerably suppressed. These trends were pronounced for the single crystal. The transition of the polycrystal became broad above 40 Oe and a characteristic tail appeared at temperatures above the peak of C/T. We suggest that the tail is induced by domains that have crystallographic disorders of the non-centrosymmetry and a higher critical magnetic field than that of the bulk. These domains became superconducting at temperatures above the peak of C/T, and then formed the tail. Both the poly- and single-crystalline LaPt3Si samples have been found to show characteristics that are in-between those of type-I and type-II superconductors.