The relationship between antiferromagnetic (AF) fluctuation and superconductivity was investigated in the La1111 series, LaFeAsO1- xFx (x = 0, 0.05, 0.08, 0.10, and 0.14) by examining nuclear relaxation rates (1/T-1) at both ambient pressure and 3.0 GPa. The results show that the critical doping level at which low-frequency AF fluctuation vanishes is around the optimally doped regime (x similar to 0.10). Although the AF fluctuation is enhanced by applying pressure in the underdoped regime (0.05 <= x < 0.10), the increase in critical transition temperature (T-c) is small, whereas Tc remarkably increases in the overdoped regime (x = 0.14), implying that the AF fluctuation is less important to the high-T-c mechanism than the density of states at the electron pocket. The x dependence of T-c at 3.0 GPa is similar to that of R1111 (R = Ce, Pr, Nd, etc.) with T-c >= 40 K at ambient pressure. The relationship between Tc and the pnictogen height or lattice constant indicates that pressure application is equivalent to full rare-earth substitution. This equivalence suggests that high T-c above 40 K is realized when the AF fluctuation is absent.