Let $X = \left(L \times [0, 1]\right) / \left(L \times \{0\}\right)$ be the closed cone over a closed smooth $d$-dimensional manifold $L^{d}$. Let $i \colon Y \hookrightarrow X$ denote the inclusion of the open cone $Y = \left(L \times [0, 1)\right) / \left(L \times \{0\}\right)$.

Let $\boldsymbol{IC}^{\bullet}_{\overline{p}}(Y)$ denote the perversity $\overline{p}$ intersection chain complex of the open cone $Y$ (using $\mathbb{Q}$-coefficients) as introduced in M. Goresky and R.D. MacPherson's paper Intersection homology II, Invent. Math. 71 (1983), 77-129, then it is well-known that the perversity $\overline{p}$ intersection homology of $Y$ can be computed to be the following cotruncation of $H_{\ast}(L)$:

$IH_{i}^{\overline{p}}(Y) \cong \mathcal{H}^{-i}(Y; \boldsymbol{IC}^{\bullet}_{\overline{p}}(Y)) \cong \begin{cases} 0, \quad i \leq d - \overline{p}(d+1), \\ H_{i-1}(L), \quad i > d - \overline{p}(d+1). \end{cases}$

**My question**:
Is there a similar formula for $\mathcal{H}^{\ast}(X; i_{\ast}\boldsymbol{IC}^{\bullet}_{\overline{p}}(Y))$, i.e. the hypercohomology of the push-forward of $\boldsymbol{IC}^{\bullet}_{\overline{p}}(Y)$ to $X$?
And if so, how can it be derived?