"quod erat demonstrandum" i.e. "Proof neads demonstration" is absolutely right with Riemann's zeta as a function of complex numbers..
In my humble opinion, we can not answer if some one asks us, "what is 3a compared to 4b?", unless we know about a and b!!
In the assumption we have made here, s > 0 or s < 1/2, we compare s as only real number where as s is a complex number involving imaginary ones about which we donot know. I can not say (-1)^0.5 > 0. Not even 10((-1)^0.5) > 0.
I am just a novice with high regards for Riemann and hence commented.
It seems from the attached doc, s may be a complex number only.