Explicit Waldspurger-type formula for level $p^2$ Shimura lifts

Let $p$ be a prime, let $f$ be the modular form in the construction, let $\widetilde{\mathscr{O}}$ and $\mathscr{O}'$ be the relevant orders, and write

$$\alpha_f=\frac{1}{2}\begin{cases}1&\text{if $f$ is not the twist of a level $p$ form},\frac{p}{p-1}&\text{if $f$ is the quadratic twist of a level $p$ form}.\end{cases}$$

Let $c_{f,\mathscr{O}'}(d)$ be the coefficient of $q^d$ in the constructed weight $3/2$ modular form, and let $\langle\,\ \rangle$ denote the relevant Petersson inner products. Explicit Shimura correspondence formula. If $d$ is an integer such that $-pd<0$ is a fundamental discriminant and $\left(\frac{d}{p}\right)=\sigma(\mathscr{O}')$, then

$$L(f,-pd,1)L(f,1)=\alpha_f\,\frac{\langle f,f\rangle}{\sqrt{pd}}\,\frac{c_{f,\mathscr{O}'}(d)^2}{\langle\mathbf e_{f,\mathscr{O}'},\mathbf e_{f,\mathscr{O}'}\rangle}.$$

This is presented as an explicit version of the preceding nonvanishing conjecture and is reported to have been verified numerically in many cases; it refines the expected relation between central $L$-values and Fourier coefficients of the Shimura-correspondent form.