The frequency and wave-vector dependent dielectric function of Al is calculated, which is formulated by the random-phase approximation. The local-field effects are not considered. An ab initio pseudopotential method is used to calculate the energy band structres of Al. A linear tetrahedron method is employed to perform the reciprocal space integration. The real and imaginary parts of the dielectric function are obtained through direct calculations. For the wave-vectors q = (0,0,0), (0.25,0,0), (0.5,0,0) and (0.75,0,($2 \pi /a$), the calculations are done for energies up to about 40 eV. We also examine the plasmon modes from the results of calculations for $\varepsilon(q, \omega$). The dependences of the plasmon frequency and the static dielectric function on the wave-vector q are compared with those obtained from the Lindhard formula. The accuracy of the calculations is tested using the Kramers-Kronig relations and the sum rules.