  We consider the partial theta function $\theta (q,z):=\sum
_{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $z\in \mathbb{C}$ is a variable and
$q\in \mathbb{C}$, $0<|q|<1$, is a parameter. Set $\alpha _0~:=~\sqrt{3}/2\pi
~=~0.2756644477\ldots$. We show that, for $n\geq 5$, for $|q|\leq 1-1/(\alpha
_0n)$ and for $k\geq n$ there exists a unique zero $\xi _k$ of $\theta (q,.)$
satisfying the inequalities $|q|^{-k+1/2}<|\xi _k|<|q|^{-k-1/2}$; all these
zeros are simple ones. The moduli of the remaining $n-1$ zeros are $\leq
|q|^{-n+1/2}$. A {\em spectral value} of $q$ is a value for which $\theta
(q,.)$ has a multiple zero. We prove the existence of the spectral values
$0.4353184958\ldots \pm i\, 0.1230440086\ldots$ for which $\theta$ has double
zeros $-5.963\ldots \pm i\, 6.104\ldots$.
