I'm looking for an entire function with the property that $f(\sqrt{n+1/2}) = 0$ for $n=0,1,2,\dots$, all of which are simple zeros and $f$ has no other zeros.
I know that such functions exist and can be written as an infinite product (Weierstrass factorization). Something like
$$ f(z) = \prod_{n=0}^\infty \left( 1 - \frac{z}{\sqrt{n+1/2}} \right) \exp\left(\frac{z}{\sqrt{n}}+\frac{z^2}{2n} \right) $$
My question is: Can this function, or another function with the same zeros, be expressed in terms of known functions? My first thought was to use $\frac{1}{\Gamma(1/2-z^2)}$, but of course this also has roots at $z=-\sqrt{n+1/2},\ n=0,1,\dots$.
Any help would be greatly appreciated.