I have a formula for the special case $p=2$ for the Hilbert symbol:
\begin{equation*} \left( \frac{u 2^n , v 2^m}{2} \right) = (-1)^{\frac{u-1}{2}\cdot\frac{v-1}{2}}(-1)^{n\frac{v^2-1}{8}}(-1)^{m\frac{u^2-1}{8}} \end{equation*} with $u 2^n, v 2^m \in \mathbb{Q}_2^\times$ and $u, v\in\mathbb{Z}^\times_2$.
Simple question: How shall I interpret $\frac{u-1}{2}$, $\frac{u^2-1}{8}$, ...? Thank you!
I found a real definition in the meanwhile:
Both expression have to be interpreted modulo 2. We note that the case $u, v \equiv 0 \bmod{2}$ is not possible since $u, v \in \mathbb{Z}_p^\times$, so we get (by an abuse of notation) \begin{equation*} \frac{u-1}{2} := \begin{cases}0 & \text{ if }u\equiv +1 \bmod{4} \\ 1 & \text{ if }u\equiv -1 \bmod{4} \end{cases} \\ \frac{u^2-1}{8} := \begin{cases}0 & \text{ if }u\equiv \pm1 \bmod{8} \\ 1 & \text{ if }u\equiv \pm5 \bmod{8} \end{cases} \, . \end{equation*}