  In recent work, Sun constructed two $q$-series, and he showed that their
limits as $q\rightarrow1$ give new derivations of the Riemann-zeta values
$\zeta(2)=\pi^2/6$ and $\zeta(4)=\pi^4/90$. Goswami extended these series to an
infinite family of $q$-series, which he analogously used to obtain new
derivations of the evaluations of $\zeta(2k)\in\mathbb{Q}\cdot\pi^{2k}$ for
every positive integer $k$. Since it is well known that
$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$, it is natural to seek further
specializations of these series which involve special values of the
$\Gamma$-function. Thanks to the theory of complex multiplication, we show that
the values of these series at all CM points $\tau$, where $q:=e^{2\pi i\tau}$,
are algebraic multiples of specific ratios of $\Gamma$-values. In particular,
classical formulas of Ramanujan allow us to explicitly evaluate these series as
algebraic multiples of powers of $\Gamma\left(\frac{1}{4}\right)^4/\pi^3$ when
$q=e^{-\pi}$, $e^{-2\pi}$.
