We report a systematic study by 75 As nuclear-quadrupole resonance in LaFeAsO1−xFx. The antiferromagnetic spin fluctuation (AFSF) found above the magnetic ordering temperature TN = 58 K for x = 0.03 persists in the regime 0.04 ≤ x ≤ 0.08 where superconductivity sets in. A domeshaped x-dependence of the superconducting transition temperature Tc is found, with the highest Tc = 27 K at x = 0.06 which is realized under significant AFSF. With increasing x further, the AFSF decreases, and so does Tc. These features resemble closely the cuprates La2−xSrxCuO4. In x = 0.06, the spin-lattice relaxation rate (1/T1) below Tc decreases exponentially down to 0.13 Tc, which unambiguously indicates that the energy gaps are fully-opened. The temperature variation of 1/T1 below Tc is rendered nonexponential for other x by impurity scattering.