By definition of the Riemann Zeta Function, $\zeta\left(\frac{1}{2}\right) = \sum_{n=1}^\infty \frac{1}{\sqrt{n}}.$ Since $\forall n \geq 1 : \frac{1}{\sqrt{n}} \geq \frac{1}{n}$, we have that for all $N \geq 1$, $\sum_{n=1}^N\frac{1}{\sqrt{n}} \geq \sum_{n=1}^N \frac{1}{n},$ but it is well known that $\lim_{N\rightarrow\infty}\sum_{n=1}^N\frac{1}{n}=\infty,$ so $\zeta\left(\frac{1}{2}\right)$ diverges by the comparison test.
In other words, $\zeta\left(\frac{1}{2}\right)$ should equal positive infinity, correct? If so, why do Maple, Mathematica, and Matlab all return a value of around $-1.4604$ when asked to numerically approximate this value? For example, see here.