We adopt a gauge-invariant definition to calculate the spin and orbital angular momenta of a so-called Ith order Laguerre-Gaussian laser.The results reveal that photons on the axis of the beam may carry an orbital angular momentum of (l - 1)h besides lh per photon.For the spin, we obtain a more reasonable expression proportional to the beam intensity instead of the gradient of the intensity as previously derived.We also discuss how to experimentally discriminate the angular momentum expressions given here and those commonly accepted in the literature.