here is the following theorem:
"Given the two functions F(s) and G(s) represented by Dirichlet series with
$L(s,f)=\sum_{n=1}^{\infty} \frac{f(n)}{n^s}$ for $\sigma>a$ and
$L(s,g)=\sum_{n=1}^{\infty} \frac{g(n)}{n^s}$ for $\sigma>b$
then in the half plane where both series converge absolutely we have
L(s,h)=L(s,f)L(s,g)=$\sum_{n=1}^{\infty} \frac{h(n)}{n^s}$ with h=f*g. (convolution of f and g)"
Here is my issue: We know that L(s,$\mu$)=1/$\zeta(s)$ and L(s,$\mu^2$)=$\zeta(s)/\zeta(2s)$ converge absolutely for Re(s)>1. Suppose I want to find L(s, $\mu * \mu^2$) for s=1. We can see that L(s,$\mu$)L(s,$\mu^2$)=1/$\zeta(2s)$ and one can easily see that 1/$\zeta(2s)$ converges absolutely for Re(s)>1/2. (so their product converges absolutely at s=1, even if L(s,$\mu$) and L(s,$\mu^2$) do not.)
Can I still apply the theorem to conclude that L(1, $\mu * \mu^2$) = 1/$\zeta(2*1)$=1/$\zeta(2)$?
Thanks!