Birch and Swinnerton-Dyer special-value formula for modular abelian varieties

Let $A_f$ be the modular abelian variety associated to a weight-$2$ newform of level $N$, and suppose that $L_{A_f}(1)\ne 0$. Let $\Omega_{A_f}$ be the real volume defined using a generator of the invariant differentials, let $c_p(A_f)$ be the arithmetic component-group order at $p$, and let $A_f^\vee$ be the dual abelian variety. The Shafarevich-Tate group of $A_f$ is denoted by ${\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}(A_f)$. Birch and Swinnerton-Dyer conjecture. One has the conjectural equality

$$\frac{L_{A_f}(1)}{\Omega_{A_f}}\stackrel{?}{=}\frac{|{\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}(A_f)|\prod_{p\mid N}c_p(A_f)}{|A_f(\mathbf{Q})|\,|A_f^\vee(\mathbf{Q})|}.$$

The formula predicts the order of the Shafarevich-Tate group from the algebraic part of the special $L$-value and the other arithmetic factors. The source states no resolution of this formula.