Need to evaluate the Laplace integral transform (definition below) of $\sqrt{t}$ using complex analysis. How is that possible?

$$\hat{f}(s)={\int}_{0}^{\mathrm{\infty}}\sqrt{t}{e}^{-st}dt$$

$$\hat{f}(s)={\int}_{0}^{\mathrm{\infty}}\sqrt{t}{e}^{-st}dt$$