Suppose $|a_{n}| \leq 1$ completely multiplicative function assuming real values. Suppose further that, $ L(s)=\sum_{n=1} \frac{a_{n}}{n^s} $ may be continued analytically to the left of $s=1$ a bit (say $Re(s)>1-\epsilon$). Can we conclude that $L(s)$ converges for $Re(s)>1-\frac{\epsilon}{2}$ ?

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