Limit involving Gamma function

$$L=\lim_{s\to 0^+}\left(\Gamma(s)-\frac{1}{2}\Gamma\left(\frac{s}{2}\right)\right)=\lim_{s\to 0^+}\frac{\Gamma(s+1)-\Gamma\left(\frac{s}{2}+1\right)}{s}\tag{1}$$ 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.