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.