The question is in the title: has a universality theorem in the sense of Voronin been proved for the Davenport-Heilbronn function, or do we expect such a theorem to hold true only for L functions that are supposed to verify the analogue of the Riemann Hypothesis?  

Thanks in advance.



Has a universality theorem been proved for the Davenport-Heilbronn L function?
<http://siauliaims.su.lt/pdfai/2008/laurincikas-08.pdf>



> 
> **Abstract.** In the paper, the joint universality in the Voronin sense for
>  Hurwitz zeta-functions with parameters $a\_1; \dots ; a\_r$ such that the system
>  $\{ \log(m + a\_j) : m = 0;1;2; \dots ; j = 1; \dots ; r \}$ is linearly independent over
>  the field of rational numbers is obtained.
> 
> 
> 


For irrational $q$, the requirements are true (see Corollary 2).


Interesting side remark:



> 
> Quote: If q is algebraic irrational, then J. W. S. Cassels proved that at least 51
>  percent of elements of the set
>  $\{ log(m + q) : m \in \mathbb{N}\_0 \}$ are linearly independent over $\mathbb{Q}$.
> 
> 
> 


