  The series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ converges for
$q\in [0,1)$, $x\in \mathbb{R}$, and defines a {\em partial theta function}.
For any fixed $q\in (0,1)$ it has infinitely many negative zeros. For $q$
taking one of the {\em spectral} values $\tilde{q}_1$, $\tilde{q}_2$, $\ldots$
(where $0.3092493386\ldots =\tilde{q}_1<\tilde{q}_2<\cdots <1$, $\lim
_{j\rightarrow \infty}\tilde{q}_j=1$) the function $\theta (q,.)$ has a double
zero $y_j$ which is the rightmost of its real zeros (the rest of them being
simple). For $q\neq \tilde{q}_j$ the partial theta function has no multiple
real zeros. We prove that $\tilde{q}_j=1-\pi /2j+(\log j)/8j^2+O(1/j^2)$ and
$y_j=-e^{\pi}e^{-(\log j)/4j+O(1/j)}$.
