What is $E[\log(Z!)]$ when $Z$ is Poisson with rate $\lambda$?

Question

Suppose $Z$ is a discrete random variable taking values in $\{0,1,2,\ldots\}$, and for $\lambda>0$, $Z$ is distributed as Poisson with rate $\lambda$. It seems a bit cumbersome to evaluate \begin{equation} E[\log(Z!)], \end{equation} where '!' denotes the factorial operation. Is there any easy way to compute this?