Muon-spin-relaxation (μSR) and magnetic susceptibility measurements have been carried out in La2−x Sr x Cu1−y Zn y O4 with x=0.13 changing y finely up to 0.10, with the aim at clarifying effects of the non-magnetic impurity Zn on the Cu-spin dynamics and superconductivity. It has been found that the y dependence of the volume fraction of the superconducting region estimated from susceptibility measurements highly correlates with that of the magnetically ordered region estimated from the μSR results. The rapid decrease in the superconducting region and the rapid increase in the magnetically ordered region by the doping of a small amount of Zn can be interpreted as follows; Zn pins the dynamical spin correlation or the dynamical stripe correlations and hence the superconductivity is destroyed around itself.