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Is there any $a, b \in \mathbb R, b \ne 0$ such that $\Gamma(a+bi)$ can be evaluated manually? (Like $\Gamma(\frac 12)$)

If there is/are, could you show me how to calculate it?

I found that $\Gamma(i)$ cannot be calculated by hand, but only can be calculated using computer.

Of course I tried

$\Gamma(i+1)=\displaystyle\int_0^\infty x^i~e^{-x}~dx$, and then use $x^i=e^{i\ln x}=\cos\ln x+i\sin\ln x$.

But still no clue..

Thanks.

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    $\begingroup$ Could anybody comment on the reason of a downvote? Where should I improve? $\endgroup$ – KYHSGeekCode Sep 22 '18 at 10:01
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    $\begingroup$ i think this is a good questions $\endgroup$ – user547564 Sep 22 '18 at 11:00
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    $\begingroup$ see this math.stackexchange.com/questions/215352/… $\endgroup$ – user547564 Sep 22 '18 at 11:14
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    $\begingroup$ @KYHSGeekCode see the following link i hope you find some clues math.stackexchange.com/questions/264034/… $\endgroup$ – user547564 Sep 22 '18 at 11:17
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    $\begingroup$ Just a thought: In Vincent Law's second link there is a claim that $\arg(\Gamma(z))$ has a closed form. If $\Re(z)=\frac{1}{2}$, then $1-z=\bar{z}$, so $$\frac{\pi}{\sin(\pi z)}=\Gamma(z)\Gamma(1-z)=\Gamma(z)\Gamma(\bar{z})=\Gamma(z)\overline{\Gamma(z)}=|\Gamma(z)|^2,$$ so (if one can find a reference for that particular claim) this might be able to give somewhat of a closed form. $\endgroup$ – Carl Schildkraut Sep 27 '18 at 6:46

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