Let
$$
D=\{z \in \mathbb{C} : 0<\Re(z)<\frac{1}{2} \text{ or } \frac{1}{2}<\Re(z)<1 \}
$$
that is the critical strip without critical line.


I have to find if the following equation, with Gamma Euler function, has any root in $D$
$$
\Gamma(z)
=
\dfrac
{\pi^z}
{\cos
\left(
\dfrac{\pi}{2} \cdot z
\right)
\cdot
2^{1-z}
}
\cdot
\dfrac
{1-2^{1-z}}
{1-2^z}
$$


Thanks.



An equation with Gamma Euler function in critical strip
Imprecise answer: if you denote by $f(z)$ the log of the ratio of the two sides, we have $f(1-z)=-f(z)$ (I assume you constructed your function in that way). One now uses an
old theorem of Hermite, unfortunately I don't remember the exact statement and reference (he states it for polynomials, but it is easily generalized) which shows that all the zeros have real part $\ge1/2$, hence by the functional equation also $\le1/2$, so are on the critical strip. Sorry to be so imprecise (same proof applies to functions such as $\Lambda(s+a)+\Lambda(s-a)$ with $a\ge1/2$ for instance, with $\Lambda(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$).


