In this article, the logarithmically complete monotonicity of some functions such as $frac{1}{[Gamma(x+1)]^{1/x}$, $frac{[Gamma(x+1)]^{1/x}}{x^alpha}$, $frac{[Gamma(x+1)]^{1/x}}{(x+1)^alpha}$ and $frac{[Gamma(x+alpha+1)]^{1/(x+alpha})}{[Gamma(x+1)^{1/x}}$ for $alpha{in}mathbb{R}$ on ($-1,infty$) or ($0,infty$) are obtained, some known results are recovered, extended and generalized. Moreover, some basic properties of the logarithmically completely monotonic functions are established.