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A Dirichlet-$L$ function is typically defined by its series, and its Euler product is a consequence of the definition. Here my approach is the other way around. I define the function $$ L_4^*(s) = \...

Let's consider the following Euler product ($s=\sigma+it)$: $$ P(s)=\prod_{p \; \text{prime}} \frac{1}{1-p^{-s}} \; e^{-p^{-s}}$$ So for $\sigma>1$, it is clear the product converges and we have: $$...