Show that $\displaystyle \zeta(s) < 0 $ for $\displaystyle 0 < s < 1 $.
The Laurent expansion of the $\displaystyle \zeta(*)$ around $\displaystyle s=1$ is...
$\displaystyle \zeta(s)= \frac{1}{s-1} + \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!}\cdot \gamma_{n}\cdot (s-1)^{n}$ (1)
... where the $\displaystyle \gamma_{n}$ are the so called 'Stieltjes constants'. For $\displaystyle s$ 'not too far' from 1 the term $\displaystyle \frac{1}{s-1}$ is dominant, so that $\displaystyle \zeta(*)$ is negative in $\displaystyle 0 < s < 1$. The nearest zero on the real axis is at $\displaystyle s=-2$...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
P.S. That's a graphich I've made about a year ago using the (1)...