= Solution
For the supplied <linear rigid rotor> spectrum, sum over every nonnegative integer $J$ and include all $2J+1$ magnetic substates:
$$
\boxed{Z_{\rm rot}=\sum_{J=0}^{\infty}(2J+1)
\exp\left[-\frac{\hbar^2J(J+1)}{2Ik_BT}\right].}
$$
It is convenient to introduce the <rotational temperature>
$$
\theta_r=\frac{\hbar^2}{2Ik_B},
$$
so that the first terms are $Z_{\rm rot}=1+3e^{-2\theta_r/T}+5e^{-6\theta_r/T}+\cdots$. Here $I$ is the <moment of inertia> and $\hbar$ is the <reduced Planck constant>, as confirmed in the original PDF. The stipulated degeneracies are used as given, without adding nuclear-spin weights or omitting selected $J$ values.
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