The effects of planar hole content, p (= x), on the static magnetic susceptibility, chi(T), of polycrystalline La2-xSrxCu1-yZnyO4 compounds were investigated over a wide range of Sr (x) and Zn (y) contents. The magnetic behavior caused by Zn was found to depend strongly on the hole content. The apparent magnetic moment induced by Zn was larger in underdoped La2-xSrxCu1-yZnyO4, decreased quite sharply around p ~ 0.19, and did not change much for further overdoping. This is interpreted in terms of the effect of the pseudogap on the Zn-induced magnetic behavior, as there is growing evidence that the pseudogap vanishes quite abruptly at p ~ 0.19 +/- 0.01. From a detailed analysis of chi(T) data the Zn-induced magnetic contribution was found to be rather complex and showed non-Curie-like features over a wide range of temperature. The observed behavior was scrutinized in terms of two scenarios (a) that of independent localized-moments and (b) low energy quasiparticle resonances associated with each Zn atom. Our study points towards the latter scenario and more generally suggests that there is a re-distribution of quasiparticle spectral weight due to Zn substitution, the features of which are greatly influenced by the presence and magnitude of the pseudogap.