A series of high-quality La2-xSrxCuO4 (LSCO) superconductor crystals in the under-doping region with x = 0.063, 0.07, 0.09, 0.10, 0.111 and 0.125 has been successfully prepared by traveling-solvent floating-zone (TSFZ) technique. The crystals are large and free of sub-grains and foreign phases. The high crystal quality has been revealed by double-crystal x-ray rocking curves and Rutherford backscattering spectrometry combined with ion-beam channeling effect. We find that the evolutions of the superconducting transition width and volume fraction as a function of carrier concentration exhibit interesting anomalies in the vicinity of some magic number doping levels such as x =1/4^2 (=0.0625) and 1/3^2 (=0.111). We argue that these behaviours are of intrinsic electronic origin.