  The main purpose of this note is to provide an algorithm for approximating
the value of the balanced $p$-adic $L$-function, as constructed in [Hsi21], at
the point $(2,1,1)$, which is lying outside of the interpolation region. The
algorithmic procedure is obtained building on the work of [FM14] and
considering finite length geodesics on the Bruhat--Tits tree for
$GL_2(\mathbb{Q}_p)$. We are interested in the case where at least one of the
Hida families is associated with an elliptic curve over the rationals and we
further restrict ourselves to the case where only one finite local sign of the
functional equation is $-1$.
