= Solution
Treat $\alpha$ as a constant and use the <canonical ensemble> identity $F=-k_BT\ln Z$. The given formal <partition function> gives $F=\alpha k_BT^3N^2/V$. The <pressure> is its negative <volume> derivative at fixed $T,N$:
$$
\boxed{P=-\left(\frac{\partial F}{\partial V}\right)_{T,N}=\alpha k_BT^3\frac{N^2}{V^2},\qquad PV^2=\alpha k_BT^3N^2.}
$$
Equivalently $P=k_BT(\partial\ln Z/\partial V)_{T,N}$. The explicit <temperature> dependence in $\ln Z$ must be retained; this is not the ideal-gas <pressure> law.
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