We have studied the effect of heat treatment and the Ge substitution in place of Si in the recently discovered heavy-fermion superconductor CePt3Si. The annealed CePt3Si exhibited nonmagnetic heavy-fermion behavior instead of the antiferromagnetism (AF) found in quenched samples. The AF state was destroyed by only about 1 at.% of Ge-substitution and may not be a stable phase. Specific-heat measurements on the annealed CePt3Si and the Ge-substituted samples revealed a large hump around 2.2 K, originally claimed as Neel temperature. Its true nature is not clarified yet but conjectured at present as a sort of quadrupolar transition rather than AF long-range order. The superconducting transition around 0.75 K was equally sharp with DeltaC(P)/gammaT(c) = 0.7 for clean quenched and annealed samples. The interplay between the 2.2 K-anomaly and the superconductivity is discussed. (C) 2004 Elsevier Ltd. All rights reserved.