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

((This is to be done by writing
$L(s)=exp(\sum_{p} \frac{a_{p}}{p^s})G(s)$, where $G(s)$ is a harmless function and by partial summation together with the bound, $\sum_{p \leq x} a_{p} \ll x^{1-\epsilon}$ )) .  
Can we conclude that $L(s)$ converges for $Re(s)>1-\frac{\epsilon}{2}$ ?
 

Regards.