We report a systematic study by (75)As nuclear-quadrupole resonance in LaFeAsO(1-x)F(x). The antiferromagnetic spin fluctuation found above the magnetic ordering temperature T(N) = 58 K for x = 0.03 persists in the regime 0.04 ≤ x ≤ 0.08, where superconductivity sets in. A dome-shaped x dependence of the superconducting transition temperature T(c) is found, with the highest T(c) = 27 K at x = 0.06, which is realized under significant antiferromagnetic spin fluctuation. With increasing x further, the antiferromagnetic spin fluctuation decreases, and so does T(c). These features resemble closely the cuprates La(2-x)Sr(x)CuO(4). In x = 0.06, the spin-lattice relaxation rate (1/T(1)) below T(c) decreases exponentially down to 0.13T(c), which unambiguously indicates that the energy gaps are fully opened. The temperature variation of 1/T(1) below T(c) is rendered nonexponential for other x by impurity scattering.