Mathematica returns the following Laurent Series for the $\wp(z;(1,1+I))$ function
Series[WeierstrassP[z, WeierstrassInvariants[{1, 1 + I}]], {z, 0, 12}]
$$\frac{1}{z^2}+\frac{\Gamma \left(\frac{1}{4}\right)^8}{5120 \pi ^2}z^2 +\frac{\Gamma \left(\frac{1}{4}\right)^{16}}{78643200 \pi ^4}z^6 +\frac{\Gamma \left(\frac{1}{4}\right)^{24}}{2617245696000 \pi ^6}z^{10} +O\left(z^{13}\right)$$
that is no terms for $z^4, z^8$, etc.
According to https://dlmf.nist.gov/23.9
$$ \wp(z)=\frac{1}{z^2}+\sum_{n=2}^{\infty}c_n z^{2n-2}. $$
that is including terms for $z^4, z^8$, etc.
Can you please explain the discrepancy? What have I done wrong?