I ask about an idea to prove this formula:


$Γ(1/2-iβ)=((\sqrt{π})/(\sqrt{\coshπβ}))\exp(-i(2ϑ(β)+βln2π+\arctan(\tanh(1/2)πβ)))$


where $ϑ(β)$ is the Riemann Siegel function.



Riemann Siegel function and gamma function
I know two proof, the first uses 
$$\cos\frac{\pi s}{2}=\frac{1}{\sqrt{2}}\sqrt{\cosh(\pi
t)}\,e^{-i\arctan(\tanh\frac{\pi t}{2})}.\qquad (1)$$
and 
$$\Gamma(\frac12+i\frac t 2)=|\Gamma(\frac14+i\frac t2)|\,e^{i(\vartheta(t)+\frac t 2\log\pi)},\qquad (2)$$


Since 
$$\Gamma(z)\Gamma(z+1/2)=2^{1-2z}\sqrt{\pi}\Gamma(2z);\quad
\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin\pi z}$$
we get
$$\Gamma(2z)=2^{2z-1}\pi^{-1/2} \Gamma(z)\frac{\pi}{\cos\pi
z}\frac{1}{\Gamma(1/2-z)}.$$
We put now $z=\frac14+i\frac t 2$, $t$ real
$$
\Gamma(\frac12+it)=2^{-\frac12+it}\frac{\pi^{\frac12}}{\cos\pi(\frac14+i\frac{t}{2})}
\frac{\Gamma(\frac14+i\frac{t}{2})}{\Gamma(\frac14-i\frac{t}{2})}.
$$
From (1) and (2) we get
$$\Gamma(1/2+it)=2^{it}\frac{\pi^{1/2}}{\sqrt{\cosh \pi t}\; e^{-i
\arctan\tanh(\pi t/2)}}\pi^{it} e^{2i\vartheta(t)}.
$$
so that
$$
\Gamma(1/2+it)=\sqrt{\frac{\pi}{\cosh\pi
t}}\exp\bigl\{i(2\vartheta(t)+t\log(2\pi)+\arctan\tanh(\pi
t/2))\bigr\}
$$


The other proof I know uses the functional equation of the zeta function.


