I've seen a really interesting result, namely :
$$\underset{s \to 0}{\lim} \Gamma(s) - \frac{\Gamma(\frac{s}{2})}{2} =-\frac{\gamma}{2}$$
This was computed by Wolfram|Alpha, but can someone give some insight on how the $\gamma$ constant appears? What manipulations can one use in cases like this, when the argument of the Gamma function $\rightarrow 0$ ?
$$L=\lim_{s\to 0^+}\left(\Gamma(s)-\frac{1}{2}\Gamma\left(\frac{s}{2}\right)\right)=\lim_{s\to 0^+}\frac{\Gamma(s+1)-\Gamma\left(\frac{s}{2}+1\right)}{s}\tag{1}$$ hence by applying De l'Hopital theorem and exploiting $\Gamma'(z)=\psi(z)\,\Gamma(z)$ we have:
$$ L = \psi(1)-\frac{1}{2}\psi\left(1\right)=\frac{\psi(1)}{2}=-\frac{\gamma}{2}\tag{2}$$ as wanted.