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Given two varieties of finite type over a finite field, what is the $L$-function of their product in terms of the $L$-function of the factors?

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    See Appendix C of Hartshorne for how to write down the zeta function using étale cohomology. Once that is established, the question reduces to the Künneth formula (See Corollary 8.23 of "Etale cohomology" by J.S. Milne.) The short answer is that you can write down the zeta function of the product if you know the zeta function of the factors. It will involve factoring both the numerator and denominator into linear factors and "combining" the roots.2011-05-10

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