This chapter turns to L-functions. It first covers motivic and cohomological L-functions. There is a well-known conjectural dictionary between cohomological cuspidal automorphic representations of GLn and pure rank n motives. The chapter briefly reviews this dictionary while recasting it in the context of strongly inner Hecke summands on the one hand and pure effective motives on the other. Afterward, the critical points for L-functions and the combinatorial lemma are explored. In particular, the chapter reviews the Rankin–Selberg L-functions. A proof of combinatorial lemma is also given. The chapter then provides the main result on special values of L-functions. It concludes with some remarks.
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