Is Riemann zeta function injective in some strips $a<\Re(s)Or more generally, are L-functions injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
 A: I think the answer is negative for the $\zeta$-function as well as for many $L$-functions and even products of them because injectivity in such strips would contradict various universality theorems. More precisely for Riemann's $\zeta$-function:
By symmetry it suffices to consider the half-strip $1/2<\sigma<1$. Voronin's universality tells us that for any compactum $K$ on this half-strip with connected complement, any non-vanishing $f(s)$ holomorphic on the interior of $K$ can be approximated arbitrarily well uniformly by a shift $\zeta(s+i\tau)$, $s=\sigma+it$, for some fixed $\tau>0$, i.e. $\forall\varepsilon>0\ \exists\tau>0: ||\zeta(s+i\tau)-f(s)||_K<\varepsilon$ (in fact, the amount of such $\tau$-s has positive lower density). Now take a sufficiently non-injective $f$.
Similar results hold for many $L$-functions and products thereof (just google "universality of L-functions" to find some surveys).
Maybe there are easier arguments against non-injectivity of the $\zeta$-function on strips inside the critical strip.
A: 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.
