A domain containing all zeros of the partial theta function

We consider the partial theta function, i.e. the sum of the bivariate series $\theta (q,z):={\sum }_{j=0}^{\mathrm{\infty }}{q}^{j(j+1)\mathrm{/}2}{z}^j$ for $q\in (0,1)$, $z\in $\mathbb{C}$$\mathrm{.}Weshowthatforanyvalueoftheparameter$q\in (0,1)$ all zeros of the function $\theta (q,\mathrm{.})$ belong to the domain ${ {\rm Re}~z<0, |{\rm Im}~z|\leq 132}$$\cup$${ {\rm Re}~z\geq 0, |z|\leq 18}$.