Paajanen's abscissa formula for normal zeta functions of class-two Lie rings

Let $L$ be a class-$2$-nilpotent Lie ring with $\operatorname{rk}(L/L')=d$, $\operatorname{rk}(Z(L))=m$, and $\operatorname{rk}(L/Z(L))=r$. Let $\alpha^{\triangleleft}$ be the abscissa of convergence of the normal-subring zeta function $\zeta^{\triangleleft}_L(s)$. Paajanen's abscissa formula.

$$\alpha^{\triangleleft}=\max_{k\in[m]}\left\{d,\frac{k(m+d-k)+1}{r+k}\right\}.$$

The source records that the lower bound for $\alpha^{\triangleleft}$ was proved by Paajanen, and the displayed equality is therefore presented as resolved rather than open.