Dirichlet series with abscissa of absolute convergence $= \frac{1}{2}$

Question

I'm trying to figure out a Dirichlet series which has its abscissa of absolute convergence $=\frac{1}{2}$. I've been trying to think about using the formula for this abscissa:

$$\sigma_a=\limsup_{n\to\infty}\frac{\log(|a_1|+|a_2|+\cdots+|a_n|)}{\lambda_n}.$$ if $\sum |a_k|$ diverges.

$$\sigma_a=\limsup_{n\to\infty}\frac{\log(|a_{n+1}|+|a_{n+2}|+\cdots)}{\lambda_n}$$ if $\sum |a_k|$ converges.

Thank you in advance.