we remark on two points. First, we take the sum only over certain Kottwitz triples which we call *stable* (Def. \[defn:stable\_Kottwitz\_triple\]). This notion of stableness has the same meaning as in the stable conjugacy (e.g. every Kottwitz triple is stable in our sense if $G^{{\mathrm{der}}}=G^{{\mathrm{sc}}}$), so is a natural condition in this context. But there is also some subtle point at $p$ (cf. Remark \[rem:Kottwitz\_triples\] and ). Secondly, the condition of “*having trivial Kottwitz invariant*” should be also taken with a grain of salt: again unless $G^{{\mathrm{der}}}=G^{{\mathrm{sc}}}$, Kottwitz invariant itself is not a notion well-defined by a (stable) Kottwitz triple alone, although its vanishing is so (cf. ).

To have a complete picture of the Hasse-Weil zeta function (and for applications to construction of Galois representations, cf. [@ScholzeShin13]), one also needs similar descriptions at bad reductions (cf. [@Rapoport05 $\S$10], [@Haines14 Conj.4.30, 4.31]). For that, we remark that we also obtain some partial results towards Thm. \[thm\_intro:Kottwitz\_formula\] for (special) parahoric levels under the assumption that $G_{{{\mathbb Q}_p}}$ is quasi-split and is tamely ramified, and together with such results, the same methods proving the above theorems (based on [@Kottwitz84b], [@LR87]) should allow the wanted descriptions, once those results in [@Kisin17] which were 