In some old notes, I found the following conjecture:

Let $\mathbb T$ denote the unit circle and let $\chi:{\mathbb N} \to {\mathbb T}$ be fully multiplicative. Then the L-series $$ L(\chi,s)=\sum_{n=1}^\infty\frac{\chi(n)}{n^s}, $$ defined for ${\rm Re}(s)>1$, cannot be extended to a holomorphic function on $\{ {\rm Re}(s)\ge 0\}$.

Who formulated this conjecture? What related results are known?

provethat some lost manuscript of Archimedes (or some private letter of Euler, or...) did not contain the full discussion already? – fedja Mar 23 '12 at 10:08