On the complex conjugate zeros of the partial theta function

Abstract: We prove that 1) for any $q\in (0,1)$, all complex conjugate pairs of zeros of the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ belong to the set ${$~Re\,$x\in (-5792.7,0),$~$|$Im\,$x|<132~}$ $\cup$ ${ ~|x|<18~}$ and 2) for any $q\in (-1,0)$, they belong to the rectangle ${$~$|$Re\,$x|< 364.2,$~$|$Im\,$x|<132~}$.