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


$$ 
L\_4^\*(s) = \prod\_{p\in P} \Bigg(1 - \frac{\chi(p)}{p^{s}}\Bigg)^{-1} 
$$
with $P=\{p\_1,p\_2,\dots\}$ the set of prime numbers but ordered differently so that the sequence $\chi(p\_1),\chi(p\_2),\dots$ is perfectly alternating. So
if $p\_{k} \equiv 1 \bmod{4}$ then $p\_{k+1}\equiv -1 \bmod{4}$. In this case, $p\_{k+1}$ would be the smallest prime $> p\_k$, equal to $-1$ modulo $4$, and not already used previously in the product.


Let's pretend that $L\_4^\*(s)$ is different from the standard function $L\_4$ with the primes ordered by value in their natural order. After all no one knows, but there is considerable empirical evidence that they are identical. Is $L\_4^\*$ analytical? Even if different from $L\_4$, is it any easier to prove that the $L\_4^\*$ product converges if $\Re(s)>\frac{1}{2}$?


In the end, does the above product define a nice $L$-function worth studying? To put it differently, would proving convergence for $L\_4^\*$ be a major milestone, or a useless result because $L\_4^\*$ does not belong to the category of "interesting functions" under GRH. The reason I am asking is because I am interested in finding a Dirichlet-$L$ function easier to study (the convergence) than the classic ones, yet featuring nice properties.



Convergence of scrambled product for Dirichlet-$L$ function with modulo 4 character
The usual Euler product converges absolutely in $\{s:\Re s>1\}$, hence no matter how you permute its factors, you get the same function in $\{s:\Re s>1\}$. This function has a unique analytic continuation to $\mathbb{C}$, which is then $L(s,\chi)$. In short, the new function you are trying to define will either not be analytic in $\mathbb{C}$, or it will not be new.


