We employed commercialized fast-scan chip-calorimeter (Flash DSC1) in assistant with Fourier Transform Infrared Spectroscopy (FTIR) and Atomic Force Microscope (AFM) to investigate the annealing behaviors of poly (L-lactic acid) with dominant alfa'-crystals crystallized in the low temperature region. The results put together various annealing behaviors observed previously in different temperature regions, and suggested that continuous crystal perfection and discontinuous melting-recrystallization are competing with each other on the transition from alfa'crystals to the more stable alfa-crystals. Both two processes fit well into the unified scenario of the conventional annealing behaviors of polymer crystals; therefore, our observations confirm that alfa'-crystals contain no intrinsic crystalline lattice difference from alfa-crystals, except for their different extents of crystal perfection resulting in the shifting of melting points. In addition, we found that fast recrystallization right after melting yields less stable crystals than those nascent crystals grown from the initial melt, confirming again the consequence of instant crystallization without enough time of crystal perfection as the extreme example in alfa'-crystals.