Choosing the lattice $L=[1, \sqrt{-2}]$, what is the value of $\wp(0)$?
The definition of the Weierstrass $\wp$ function for the lattice $L$ is $$ \wp(z;L) = \frac{1}{z^2} + \sum_{\omega \in L - \{ 0 \}} \bigg(\frac{1}{(z - \omega)^2} - \frac{1}{\omega^2}\bigg) \tag{w} \label{w} .$$
My attempts
I tried to plug $z=0$, but the resulting series does not make sense. I thought about the correct way to define the value of the $\wp$ function for the lattice $L$ at $z=0$. But I couldn't figure that out either.
Background
In Cox's book "primes of the form x^2 + ny^2", according to Theorem 10.14, if $\alpha \in \Bbb C - \Bbb Z$, and $L$ is a lattice on the complex plane, then the condition $$\alpha L \subset L$$ implies that $\wp(\alpha z)$ is equal to a rational function of $\wp(z)$, i.e. $\wp(\alpha z) = \frac{A(\wp(z))}{B(\wp(z))}$, and degrees satisfy $\deg(A(x)) = \deg(B(x)) + 1 = N(\alpha)$.
So for the lattice we have chosen, and $\alpha = \sqrt{-2}$, we deduce that $\wp(\sqrt{-2} z) = \frac{A(\wp(z))}{B(\wp(z))}$.
Since $N(\sqrt{-2}) = 2$, in this case, the theorem tells us that $\deg(A(x)) = \deg(B(x)) + 1 = 2$.
By dividing $B(x)$ by $A(x)$, we can write this as
\begin{equation} \wp(\sqrt{-2} z) = a\wp(z) + b + \frac{1}{c \wp(z) +d} \tag{10.21}\label{10.21} \end{equation}
where $a$ and $c$ are non-zero complex numbers.Then the book, on page 194, says
Now that the constants $a$ and $b$ appearing in \eqref{10.21} are the unique constants such that $\wp(\sqrt{-2}z) - a \wp(z) - b$ is zero when $z = 0$.
This sentence is a bit hard for me to understand.
First, $z=0$ is assumed for the weirstrass function, but I do not know how to evaluate the weirstrass $\wp$ function, namely the series \eqref{w} at $0$. Directly substituting $z=0$ is certainly not valid, since ${1 \over z^2} = \frac{1}{0^2}$ is not defined.
Second, assuming the value $\wp(0)$ is well defined and found, \eqref{10.21} can be written as $\wp(0) = a \wp(0) + b + \frac{1}{ c \wp(0) + d}$. Even with this assumption, I am still puzzled by the quoted statement, i.e. $\wp(0) - a \wp(0) - b = 0$ for the constants $a$ and $b$ appearing in \eqref{10.21}, and constants $a$ and $b$ are the only values for which the equation holds.
Thanks in advance to anyone who can explain this. (My background is lacking, so it would be nice if you could provide the proposition used so that I can refer to it)