Approximations of the balanced triple product $p$-adic $L$-function

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 $-1$.