You can take the limit $\nu\rightarrow -3/2$ of the expression from Gradshteyn
$$Q_\nu^0(z)=\frac{ \Gamma \left(\frac{1}{2}\right) \Gamma (\nu+1)\, _2F_1\left(\frac{\nu}{2}+1,\frac{\nu}{2}+\frac{1}{2};\nu+\frac{3}{2};\frac{1}{z^2}\right)}{2^{\nu+1}z^{\nu+1} \Gamma \left(\nu+\frac{3}{2}\right)},$$
which gives
$$Q_{-3/2}^0(z)=\frac{\pi  \, _2F_1\left(\frac{3}{4},\frac{5}{4};2;\frac{1}{z^2}\right)}{4 \sqrt{2} z^{3/2}}.$$

Alternatively, you can evaluate $Q^0_{-3/2}(z)$ as the real part of Mathematica's LegendreQ[-3/2,z].