Let $\vec{a}=2 \hat{\imath}+\alpha \hat{\jmath}+\hat{k}, \dot{b}=-\hat{\imath}+\hat{k}, \vec{c}=\beta \hat{\jmath}-\hat{k}$, where $\alpha$ and $\beta$ are integers and $\alpha \beta=-6$. Let the values of the ordered pair ( $\alpha, \beta$ ) for which the area of the parallelogram of diagonals $\vec{a}+\vec{b}$ and $\vec{b}+\vec{c}$ is $\frac{\sqrt{21}}{2}$, be $\left(\alpha_1, \beta_1\right)$ and $\left(\alpha_2, \beta_2\right)$. Then $\alpha_1^2+\beta_1^2-\alpha_2 \beta_2$ is equal to