I am looking for a closed form for $$B(1-z\omega,1-z\omega^2)$$ where $B(z_1,z_2)$ is the Beta function, $\omega$ is a cube root of unity and $\Re( 1-z\omega )>0$, $\Re( 1-z\omega^2 )>0$
I tried as follows: We know that if $\omega$ is a cube root of unity then $\omega^3=1$ and $1+\omega+\omega^2=0$
By definition of the Beta function $$B(1-z\omega,1-z\omega^2)=\int_{0}^{1} x^{-z\omega}(1-x)^{-z\omega^2} dx $$ So we can write using $1+\omega+\omega^2=0$ $$B(1-z\omega,1-z\omega^2)=\int_{0}^{1} x^{-z\omega}(1-x)^{z(1+\omega)} dx $$ So we get $$B(1-z\omega,1-z\omega^2)=\int_{0}^{1} (1-x)^{z}\left(\frac{1-x}{x}\right)^{z\omega} dx $$ Any help would definitely be appreciated. Thank you!