Is it possible to find some of Gamma's non-integer values by using some formulas such as: $$\Gamma(x)\Gamma(1-x)={\pi \over sin(\pi x)}$$
I know that the only known value that When $x=1-x$ and hence we can determine $\Gamma (\frac12)=\sqrt{\pi}$ because of the special case $\Gamma(\frac12)=\Gamma(1-\frac12)$ and by this value we can find all half-integers values by using Gamma property $\Gamma(x+1)=x\Gamma(x)$.
But we can note that there are infinite number of real solutions for$\Gamma(x)=\Gamma(1-x)$ which could help to find some real values.
For example let $\alpha \approx0.026042634$
So if we let $\Gamma(\alpha)=\Gamma(1-\alpha)$ it will not make difference because $\Gamma(\alpha) \neq \Gamma(1-\alpha)$
but actually, I can graphically and numerically check that $\Gamma(\alpha+4)=\Gamma(1-(\alpha+4))\approx 6.20011657$
and by using the reflection formula I have,
$$\Gamma(\alpha+4)^2={\pi \over sin(\pi \alpha)}$$ or $$\Gamma(\alpha)^2={\pi \over [\alpha(\alpha+1)(\alpha+2)(\alpha+3)]^2sin(\pi \alpha)}$$
Now the problem is that I should use an other formula or useful relation to find the value of $\alpha$.
I tried to use the product of sin $Sin(\pi x)=x\prod_{n=1}^{\infty} (1-\frac{x^2}{n^2})$ but it seems that this formula will not help.
So is there a hope to find values for Gamma's non-integer numbers?
I heard that $\Gamma(\frac13)$ and $\Gamma(\frac14)$ has been proven to be transcendental which made me feel that finding such values is just wasting of time for the mathematicians who are interested in it.