设 $\left(\boldsymbol{\alpha}_1, \boldsymbol{\alpha}_2, \boldsymbol{\alpha}_3\right)=\left(\begin{array}{lll}1 & 1 & 1 \\ 1 & 2 & 3 \\ 2 & 3 & a\end{array}\right)$ ，则 $\operatorname{det}\left(\boldsymbol{\alpha}_1, \boldsymbol{\alpha}_2, \boldsymbol{\alpha}_3\right)=$ ________ , 当 $a=$ ________ 时，向量组 $\boldsymbol{\alpha}_1, \boldsymbol{\alpha}_2, \boldsymbol{\alpha}_3$ 线性相关．