3. Let the complex numbers be
$$
z_{1}=\sin \alpha+2 \mathrm{i}, z_{2}=1+\cos \alpha \cdot \mathrm{i}(\alpha \in \mathbf{R}) \text {. }
$$

Then the minimum value of $f=\frac{13-\left|z_{1}+\mathrm{i} z_{2}\right|^{2}}{\left|z_{1}-\mathrm{i} z_{2}\right|}$ is . $\qquad$