Particular values of the gamma function

Example

Here are some values of the gamma function.

Checks. The first row is the gamma function extends the factorial. The value at $\tfrac12$ is Gamma of one half. The functional equation $\Gamma(z + 1) = z\Gamma(z)$ gives the others: $\Gamma(\tfrac52) = \tfrac32 \cdot \tfrac12 \sqrt{\pi} = \tfrac34\sqrt\pi$, and $\Gamma(-\tfrac12) = \Gamma(\tfrac12)/(-\tfrac12) = -2\sqrt\pi$, and $\Gamma(-\tfrac32) = \Gamma(-\tfrac12)/(-\tfrac32) = \tfrac43\sqrt\pi$. For $n = 2$ the general half-integer formula gives $\frac{24}{16 \cdot 2}\sqrt{\pi} = \tfrac34\sqrt{\pi}$, in agreement.

Values at other rational points are less explicit, but the reflection formula gives products: $$\Gamma\!\left(\tfrac13\right)\Gamma\!\left(\tfrac23\right) = \frac{\pi}{\sin(\pi/3)} = \frac{2\pi}{\sqrt3}, \qquad \Gamma\!\left(\tfrac14\right)\Gamma\!\left(\tfrac34\right) = \frac{\pi}{\sin(\pi/4)} = \pi\sqrt2.$$