The Legendre family of elliptic curves is defined by \begin{equation} y^2=x(x-1)(x-\lambda). \end{equation} The integration of the algebraic one-form $dx/y$ (up to a nonzero multiple) is given by the hypergeometric series \begin{equation} F(\frac{1}{2},\frac{1}{2},1;\lambda)=\sum_{j=0}^{\infty} \binom{-\frac{1}{2}}{j}^2 \lambda^j. \end{equation} From the theory of Dwork, we can counts the points of elliptic curves using this period. More precisely, for an odd prime number $p$, we define a new power series \begin{equation} U(\lambda)=F(\frac{1}{2},\frac{1}{2},1;\lambda)/F(\frac{1}{2},\frac{1}{2},1;\lambda^p). \end{equation} Dwork showed that $U(t)$ can be analytically continued to a function defined in the $p$-adic radius $\geq 1$. While if $\lambda_0 \in\mathbb{F}_p-\{0,1\}$, let $1-a(\lambda_0)T+pT^2$ be the numerator of the zeta function of the elliptic curve $y^2=x(x-1)(x-\lambda_0)$ defined over $\mathbb{F}_p$. Then we have \begin{equation} 1-a(\lambda_0)T+pT^2=(1-U(\text{Teich}(\lambda_0))~T)(1-(p/U(\text{Teich}(\lambda_0)))~T), \end{equation} which implies $$a(\lambda_0)=U(\text{Teich}(\lambda_0))+(p/U(\text{Teich}(\lambda_0)))$$ Here $\text{Teich}(\lambda_0)$ is the Teichmuller representative of $\lambda_0$.
Use Mathematica, it is very easy to evaluate the first $1000$ terms of the power series $U(\lambda)$. The choose a $\lambda_0$, we can also compute the $p$-adic expansion of its Teichmuller representative, say to the order $p^{50}$. Then I plug $\text{Teich}(\lambda_0)+O(p^{51})$ into $U(\text{Teich}(\lambda_0))+(p/U(\text{Teich}(\lambda_0)))+O(\lambda^{1001})$, it does not converge to $a(\lambda_0)$ $p$-adicly. I have tried several prime numbers.
Does anyone know why? Do I understand the theory incorrectly?