Is there a known example of a Dirichlet L series having at least one multiple root?

Question

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.

Answer 1

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).