We consider induced representations $\operatorname {\mathrm {Ind}}_{\mathrm {P}(F)}^{\operatorname {\mathrm {G}}(F)} \pi $, where $\mathrm {P}$ is a maximal parabolic subgroup of a reductive group $\operatorname {\mathrm {G}}$ over a p-adic field F, and $(\pi , V)$ is a unitary supercuspidal representation of $\operatorname {\mathrm {M}}(F)$, $\operatorname {\mathrm {M}}$ being some Levi subgroup of $\mathrm {P}$. Imposing a certain ‘Heisenberg parabolic subgroup’ assumption on $\mathrm {P}$, we apply the method of Goldberg, Shahidi and Spallone to obtain an expression for a certain constant $R(\tilde {\pi })$, which captures the residue of a family $s \mapsto A(s, \pi , w_0)$ of intertwining operators associated to this situation, in terms of harmonic analysis on the twisted Levi subgroup $\tilde {\operatorname {\mathrm {M}}}(F) := \operatorname {\mathrm {M}}(F) w_0$. For $\operatorname {\mathrm {G}}$ absolutely almost simple and simply connected of type $G_2$ or $D_4$ (resp., $B_3$), and $\mathrm {P}$ satisfying the ‘Heisenberg’ condition, if the central character of $\pi $ is nontrivial (resp., trivial) on $\operatorname {\mathrm {A}}_{\operatorname {\mathrm {M}}}(F)$, where $\operatorname {\mathrm {A}}_{\operatorname {\mathrm {M}}}$ is the connected centre of $\operatorname {\mathrm {M}}$, our formula for $R(\tilde {\pi })$ can be rewritten in terms of the Langlands parameter of $\pi $, in the spirit of a prediction of Arthur. For the same collection of $\operatorname {\mathrm {G}}$ and $\mathrm {P}$, when these central character conditions are not satisfied, Arthur’s prediction combined with our formula for $R(\tilde {\pi })$ suggests a harmonic analytic formula for a product of one or two $\gamma $-factors associated to the situation.