For any pointed space X, the mod p Lannes-Zarati homomorphism φsH˜⁎X">⁎ is considered as a graded associated version of the mod p Hurewicz map h⁎:π⁎(QX)→H⁎QX">⁎⁎⁎ in the -term of Adams spectral sequence. Therefore, the study of the mod p Lannes-Zarati homomorphism plays an important role in the determination of the image of the mod p Hurewicz map. In this paper, we investigate the behavior of φs=φsFp=φsH˜⁎S0,s≤2">⁎, for an odd prime p.