Bosonic-string critical dimension from ghost descendants ($L_2{L}_{-2}|\Omega ⟩=(2{\alpha }_0^2+D/2-17)|\Omega ⟩$)

For the FP ghost vacuum annihilated by $b_{n\ge 0}$ and $c_{n>0}$, the FP ghost level-one and level-two descendant coefficients are $-2$ and $-17$. Matter contributes ${\alpha }_0^2$ and $2{\alpha }_0^2+D/2$. Virasoro closure from BRST nilpotence at level one fixes the total highest weight to ${\alpha }_0^2/2-1$; level two then requires $D=26$. Imposing ground-state BRST closure gives ${\alpha }_0^2=2$ and the open bosonic tachyon. Interpret the coefficients with the stated ghost zero-mode pairing.