With Bayes' rule, and $s \in \{\pm 1 \}$ for $x\in \{ 0,1 \}$, we find

$$P[s=+1|y] = \frac{p(y|s=+1)}{p(y)} P[s=+1]$$ and channel output PDF $p(y|s)$ conditioned on the transmitted symbol $s$ we obtain $$\underbrace{L(s|y)}_{\text{a posteriori L-value}} = \ln\frac{p(y|s=+1) P[s=+1]}{p(y|s=-1) P[s=-1]} = \underbrace{\ln\frac{P[s=+1]}{P[s=-1]}}_{\text{a priori L-value } L_A(s)} + \underbrace{\ln\frac{p(y|s=+1)}{p(y|s=-1)}}_{\text{channel L-value } L_{ch}(s|y)} = L_A(s) + L_{ch}(s|y).$$ The channel L-value tells us what we learn about the transmitted symbol $s$ based on the channel observation.

For an AWGN channel with zero mean and variance $\sigma^2$ (per real-/imag-component, i.e., per I-/Q-channel) the channel L-value computes as

$$L_{ch}(s|y) = \ln\frac{\exp(-\frac{(y-1)^2}{2\sigma^2})}{\exp(-\frac{(y+1)^2}{2\sigma^2})} = \frac{2}{\sigma^2}y.$$