Is there a way to realize the gamma function intuitively? My first (and probably correct) guess is no, because, for example, $\Gamma(\frac 12)=\sqrt{\pi}$ doesn't make any intuitive sense at all. Also, to me it seems like negative integers would be more likely candidates to have defined values than fractions, but negative integers aren't defined with the gamma function. I saw this question which pointed out "Euler's reflection formula":

$$\Gamma(z)\Gamma(1-z)=\frac {\pi}{\sin(\pi z)}$$

This may offer some inspiration, but most likely not. Thank you for any responses to this question!

**Edit:** I saw the marked duplicate to this question. While it is good, I think it focuses more on the mathematical rigor than any intuition (when I say this I mean something that could be taught to a first or second year calculus student) that may exist.