Hyperelliptic values of the Gamma function

We determine new particular values of the Gamma function for specific rational arguments in the interval (0,1). Introducing hyperelliptic generalisations of the trigonometric functions, we obtain a halving formula for specific rational arguments. Then we define a family of functions, which encompasses the exponential function. We present an overlooked integral definition by Gauss and explore a few general formulas for the Gamma function. Finally, we apply the new values of the Gamma function to compute certain singular values of the complete elliptic integral and some arithmetic-geometric-means and give the area of certain superellipses.