Thus we obtain: $$\begin{aligned}
\langle {\cal D}^{(q)}(z) {\cal D}^{(q-n)}(w) \rangle 
= (-1)^{-q+n-1}~N_q~N_{q-n}{\cal J}_n^{(q)}~,
\qquad {\cal J}_n^{(q)} \equiv \sqrt{\frac{\pi}{\alpha}}\int_{-\infty}^{+\infty}
\frac{d\omega~e^{-\omega^2 
\lambda}~(\omega+i\epsilon)^{n}}{(\omega^2 + \epsilon^2)^{q+1}}\end{aligned}$$ where $\lambda \equiv 2{\rm ln}(|z-w|/a)^2$, and $\alpha \equiv {\rm ln}(L/a)^2$.

Below, for definiteness, we shall be interested in the case $2<q<3$, in which the relevant correlators are given by $n=0,1,2$. One has: $$\begin{aligned}
\label{defj} 
&~&{\cal J}_{0}^{(q)}~=\sqrt{\frac{\pi}{\alpha}}~\epsilon^{-2q-1}~f_q (\epsilon^2 \lambda)~;
\nonumber \\
&~& f_q(\xi) =\sqrt{\pi}\frac{\Gamma (\frac{1}{2}+q)}{\Gamma (1 + q)}~F(\frac{1}{2},~\frac{1}{2}-q~;\xi)
+ ~\xi^{\frac{1}{2}+q}~\Gamma(-\frac{1}{2}-q)~F(1+q,~\frac{3}{2}+q~;\xi) 
\nonumber \\
&~&{\cal J}_{1}^{(q)}~=~i\epsilon{\cal J}_{0}^{(q)}~, \nonumber \\
&~& {\cal J}_{2}^{(q)}=-2\epsilon^2~{\cal J}_0^{(q)} + {\cal J}_0^{(q-1)}=-\frac{\partial}{\partial \lambda}{\cal J}_0^{(q)}-\epsilon^2{\cal J}_0^{(q)}\end{aligned}$$ where $F(a,b;z)=1 + \frac{a}{b}\frac{z}{1!}+ \frac{a(a+1)}{b(b+1)}\frac{z^2}{2!} + \dots $ is the degenerate (confluent) hypergeometric function.\
Thus, the form of the algebra away from the fixed point (‘[ *off-shell form*]{}’), i.e. for $\epsilon^2 \ne 0$, is: $$\begin{aligned}
\label{offshellalg}   
&~& \langle {\cal D}^{(q)}(z) {\cal D}^{(q)}(0) \rangle 
= {\tilde N}_q^2~\sqrt{\frac{\pi}{\xi_0}}~\left(f_q(2\xi_0)
(\frac{\alpha}{\xi_0})^q + 2~f'_q(2\xi_0)(\frac{\alpha}{\xi_0})^{q-1}~
{\rm ln}(|z/L|^2) + \right.\nonumber \\ 
&~&  \left.+ \frac{1}{2}~f''_q(2\xi_0)(\frac{\alpha}{\xi_0})^{q-2}~
4{\rm ln}^2(|z/L|^2)
+ {\cal O}(\alpha^{q-3})\right)~, \nonumber \\
&~&  \epsilon~q~\langle {\cal D}^{(q)}(z) {\cal D}^{(q-1)}(0) \rangle =
{\tilde N}_q^2\sqrt{\frac{\pi}{\xi_0}}~\left(f_q(2\xi_0)
(\frac{\alpha}{\xi_0})^{q-1} + \right. \nonumber \\
&~& \left. + 2~f'_q(2\xi_0)(\frac{\alpha}{\xi_0})^{q-2}~
{\rm ln}(|z/L|^2) + {\cal O}(\alpha^{q-3})\right)~, \nonumber \\
&~&  \epsilon^2~q^2~\langle {\cal D}^{(q-1)}(z) {\cal D}^{(q-1)}(0)\rangle
= {\tilde N}_q^2~\sqrt{\frac{\pi}{\xi_0}}~f_{q-1}(2\xi_0)
(\frac{\alpha}{\xi_0})^{q-2} + {\cal O}(\alpha^{q-3})
, \nonumber \\
&~&  \epsilon^2~q~(q-1)~\langle {\cal D}^{(q)}(z) {\cal D}^{(q-2)}(0)\rangle
