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Could anyone point me to a reference where the formal group law of (topological or motivic) K-theory is computed in as much detail as possible?

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Well, I don't think there can be much detail here: the first Chern class of a line bundle in topological K-theory is defined by $c_1(L)=\pm1\pm[L]$; hence $c_1(L\otimes L')=c_1(L)+c_1(L')\pm c_1(L)c_1(L')$ — so the FGL is just $F(u,v)=u+v\pm uv$ (sign convention varies).

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