I should solve $\pi^{n/2} / \Gamma(n/2 + 1) = 1$. Therefore, I need to know other forms of $\Gamma(n/2)$ or $(n/2)!$. I have already checked the Mathematica and MathWorld, very well. But unfortunately, I had not any progress so far.
The best way that I have done is to find the most similar statistical distribution, which is a mixture of Levy distribution and uniform distribution.
Then I tried to fit the best curve to recognize the parameters. Then I changed the parameters with their rationalized one -I mean one integer divided by another integer-, and replaced them in mixture distribution and then I fully simplified the final expression completely.At the end, I verified it to measure the residuals and I repeated the above procedure again and again to find the better and better result.
However, it is a approximately solution, but I need a exact one.
You could get a nice approximation of the solution.
Let $m=\frac n2$ which means that you want to solve the equation $$m! =\pi ^m$$
If you look at this question of mine, you will see a magnificent approximation by @robjohn.
Applied to you case $(a=\pi,k=0)$ $$m=-\frac{\log (2 \pi^2 )}{2 W\left(-\frac{\log (2 \pi^2 )}{2 e \pi}\right)}-\frac{1}{2}$$ that is to say $$n=-\frac{\log (2 \pi^2 )}{ W\left(-\frac{\log (2 \pi^2 )}{2 e \pi}\right)}-1$$ which is $\approx 12.748$ (relative error of $0.12$%).