For abelian varieties over $\mathbb{Q}$ $\mathscr{A}$ and $\mathscr{A}'$, if derived categories $D(\mathscr{A})$ and $D(\mathscr{A}')$ are equivalent then L-functions are same $L(s,\mathscr{A})=L(s,\mathscr{A}')$.

Is this true for any smooth projective varieties? In other words, for smooth projective varieties $X$ and $Y$, $D(X)\simeq D(Y)$ then $L(s,H^i(X))=L(s,H^i(Y))$ ?

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