Bayesian server error rate with a Gamma prior

You model the number of server errors per hour as a Poisson process with unknown average rate $\lambda$ (errors per hour). Before collecting data your belief about $\lambda$ is a $\text{Gamma}(\alpha=3, \beta=2)$ prior, where $\alpha$ is the shape and $\beta$ is the rate. You then watch the server for 5 hours and record a total of 14 errors.

Using Bayesian updating, what is the posterior mean estimate of the error rate $\lambda$?