Recall the definition

Using gamma function , we have:

$\begin{aligned} \Gamma (s) & = (s-1)! \\ \implies \left( - \frac 32 \right)! & = \Gamma \left( - \frac 12 \right) \end{aligned}$

Using Euler's reflection formula:

$\begin{aligned} \Gamma (z) \Gamma (1-z) & = \frac \pi{\sin (\pi z)} & \small \color{#3D99F6}{\text{Putting }z = \frac 12} \\ \Gamma \left( \frac 12 \right) \Gamma \left( 1 - \frac 12 \right) & = \frac \pi{\sin \left( \frac 12 \pi \right)} \\ \Gamma^2 \left(\frac 12 \right) & = \pi \\ \implies \Gamma \left(\frac 12 \right) & = \sqrt{\pi} \end{aligned}$

Putting $z = \frac 32$ ,

$\begin{aligned} \Gamma \left( \frac 32 \right) \Gamma \left( 1 - \frac 32 \right) & = \frac \pi{\sin \left( \frac 32 \pi \right)} \\ \color{#3D99F6}{\Gamma \left( \frac 32 \right)} \Gamma \left(- \frac 12 \right) & = - \pi & \small \color{#3D99F6}{\text{Note that }\Gamma (s+1) = s \Gamma (s)} \\ \color{#3D99F6}{\frac 12 \Gamma \left( \frac 12 \right)} \Gamma \left(- \frac 12 \right) & = - \pi \\ \color{#3D99F6}{\frac \pi 2} \Gamma \left(- \frac 12 \right) & = - \pi \\ \implies \Gamma \left(- \frac 12 \right) & = \boxed{- 2 \sqrt \pi} \end{aligned}$