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The Davenport-Heilbronn function doesn't have a Dirichlet series since its a linear combination of two L functions. I mention that because its a common myth that this function is an L function.

gymbvghjkgkjkhgfkl
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crow
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  • $a , L(s,\chi_5) +\overline{a} , L(s,\overline{\chi_5})$ is a Dirichlet series, it belongs to the extended Selberg class (it has a functional equation) but it doesn't have an Euler product (its coefficients are not multiplicative) so it is not a L-function. – reuns Mar 14 '17 at 21:12

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No, and it is conjectured that every zero is simple. There are a variety of partial proofs towards this, proving results of the flavor "$X$ percent of zeroes up to height $Y$ are simple."

This is something specific to Dirichlet $L$-functions, as other $L$-functions are known to have higher order zeroes. For instance, Hasse-Weil $L$-functions often have multiple order roots at $s = 1$. (This is a fundamental aspect of the Birch and Swinnerton-Dyer Conjecture).

davidlowryduda
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  • thank you @mixedmath Do you know of any results that say for instance, every root up to height L is simple? Instead of some percentage of every root up to some height – crow Mar 26 '17 at 19:26