I found this when playing with complex function plotter (just enter zeta(z)*gamma(z/2+1)).
This eliminates all trivial zeroes of Riemann zeta function, does not create any new singularity, and also the resulting graph (especially along the negative real axis) "looks really smooth".
Thus, I think there must be something special. But within my search for this, little is obtained. The closest match I've found is at Taylor coefficients for $\zeta(x)\Gamma(x)$ where they're multiplied together but no coefficient on argument is present.
Wolframalpha doesn't give valuable idea either. It even refuses to find the local minima for $\frac 1 {\zeta(x)\Gamma(x/2+1)}$.