Mazur–Tate–Teitelbaum p-adic BSD conjecture for elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve with analytic rank $r_{\mathrm{an}}(E/\mathbb{Q})=1$. Let $\mathcal{L}_{p,\alpha}(t)$ be the $\alpha$-adic $p$-adic $L$-function, let $\alpha$ be the unit root, and let $\langle\cdot,\cdot\rangle_{p,\alpha,L}$ be the associated $p$-adic height pairing. Let $P$ generate $E(\mathbb{Q})_{/\mathrm{tors}}$, and let $\chi_{\mathrm{cyc}}$ be the $p$-adic cyclotomic character.

Mazur–Tate–Teitelbaum's p-adic BSD conjecture. One should have

$$\mathcal{L}_{p,\alpha}^{\prime}(0)=\frac{1}{\log_p(\chi_{\mathrm{cyc}}(\gamma))}\left(1-\frac{1}{\alpha}\right)^2\frac{\#\Sha(E/\mathbb{Q})\cdot\operatorname{Tam}(E/\mathbb{Q})\cdot\langle P,P\rangle_{p,\alpha,L}}{(\#E(\mathbb{Q})_{\mathrm{tors}})^2}.$$

The paper uses this conjecture in its $p$-part under hypotheses from its main theorem; the full asserted equality is presented as a conjectural $p$-adic analogue of BSD.