Estimating an Exponential Survival Probability

Let $X_1,\ldots,X_6$ be independent and identically distributed exponential random variables with unknown rate parameter $\lambda>0$, so that $$ f(x;\lambda)=\lambda e^{-\lambda x},\qquad x>0. $$ We wish to estimate the survival probability $$ \theta(\lambda)=\mathbb P_\lambda(X>3)=e^{-3\lambda}. $$ Suppose that the observed sum of the six random variables is $$ T=\sum_{i=1}^{6}X_i=12. $$ Let $\widehat{\theta}(T)$ be the uniformly minimum-variance unbiased estimator (UMVUE) of $\theta(\lambda)$. If $\widehat{\theta}(12)$ is expressed as a reduced fraction $a/b$, find the denominator $b$.