Zeros of linear twists of $L$-functions outside the critical strip

In this note we investigate the existence of zeros of linear twists of $L$-functions outside of the critical strip. In particular, we show that the Lerch zeta function $L(λ,α,s)$ has infinitely many zeros for $1<σ<1+η$, for any $η>0$, when $λ$ is irrational and $α$ is rational. This settles the question on the existence of zeros of the Lerch zeta functions for $σ>1$.

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