  We prove that there exists a one-parameter meromorphic solution $u(\tau)$
vanishing at $\tau=0$ of the degenerate third Painlev\'e equation,
\begin{equation*} u^{\prime \prime}(\tau) \! = \!
\frac{(u^{\prime}(\tau))^{2}}{u(\tau)} \! - \! \frac{u^{\prime}(\tau)}{\tau} \!
+ \! \frac{1}{\tau} \! \left(-8 \varepsilon (u(\tau))^{2} \! + \! 2ab \right)
\! + \! \frac{b^{2}}{u(\tau)},\qquad \varepsilon=\pm1,\quad\varepsilon b>0,
\end{equation*} for formal monodromy parameter $a=\pm i/2$. We study
number-theoretic properties of the coefficients of the Taylor-series expansion
of $u(\tau)$ at $\tau=0$ and its asymptotic behaviour as $\tau\to+\infty$.
These asymptotics are visualized for generic initial data.
