Or more generally, are L-functions injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?



Is Riemann zeta function injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
The Riemann zeta function function is not injective in any strip $a<\Re(s)<b$ with $\frac{1}{2}<a<b$. 


For $b\leq 1$ this follows, for example, from Theorem 11.10 in Titchmarsh: The theory of the Riemann zeta function (see also the remarks after the theorem in the book). 
The theorem (which is probably due to Bohr) implies that for any nonzero complex number $c$ and for any sufficiently large positive number $T$, there are $\gg T$ points in the rectangle $a<\Re(s)<b$ and $0<\Im(s)<T$ such that $\zeta(s)=c$. The implied constant here depends on $a,b,c$.


Alternatively, still for $b\leq 1$, the same conclusion follows from the simpler Theorem 11.9 (which states that the values $\log(\sigma+it)$ are dense in $\mathbb{C}$ for any $\frac{1}{2}<\sigma<1$) in conjunction with the open mapping theorem.


For $b>1$, we can employ Theorem 11.8 (A) in a similar fashion to reach the required conclusion.


