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

Regards.