I assume you have already found <a href="http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html">this page</a>?  It seems that, from the definition
$$ H(x) = \frac{\Psi(1-x/2)-\Psi(1/2-x/2)}{2\Gamma(1-x)} $$
the various properties you are interested in should be straightforward to prove.