Yes, $h^2(X)$ decomposes as $\mathbb{L}^\rho \oplus t(X)$ where $\rho$ is the Picard number of $X$ and you want to consider the submotive $t(X)$. Do you have references for the cases you mention? Are there such examples for $\rho<20$? Thanks

Lol, you are right. I forgot the -1 in my own computation Thanks

Merci François! Do you have an idea of how to compute the action of complex conjugation on $H^{1,1}$ of a K3 surface, at least in some examples?

Dear David, thanks for your answer! What do you use to compute $L_\infty(M^\vee, 1-s)$? I guess one has $M^\vee=M(3)$ by Poincaré duality, so $L_\infty(M^\vee, 1-s)=L_\infty(M, 4-s)$. But then $s=3$ would also be a critical value. Where is the mistake?