I am reading about L-functions of Elliptic curves from Milne's notes. The definition of the L-function of an Elliptic curve defined over $\mathbb Q$ is $$L(E,s)=\prod_{p\text{ good}} (1-a_pp^{-s}+p^{1-2s})^{-1}\prod_{p\text{ bad}}(1-a_pp^{-s})^{-1}$$ where for good primes $p$, we have that $a_p=\begin{cases}\#\tilde{E}(\mathbb F_p)&\text{good reduction at }p\\0 &\text{ additive reduction at }p\\1& \text{multiplicative split reduction at } p\\-1&\text{multiplicative non-split reduction at }p\end{cases}$
I am unable to understand the rationale behind the choice of the coefficients of the $p$-factors of the $L$-functions. The only possible explanation that I have found out till now is that the functional equation is satisfied because of the choice.
I would prefer a clearer and more natural explanation for the coefficients. Some historical background is also appreciated.
This is the so-called Hasse-Weil L-function of an elliptic curve over $\mathbb{Q}$, which is defined as the product over its local $L$-series: $$ L(E,s):=\prod_p L_p(E,s)^ {-1}, $$ related to the Hasse-Weil Zeta function, i.e., by $$ Z_{E,\mathbb{Q}}(s)=\frac{\zeta(s)\zeta(s-1)}{L(E,s)} $$ Regarding your question: there is really a lot of mathematics going on behind this definition. This has been explained on this site and at Mathoverflow already quite good. A starting point could be this question:
$L$-functions of elliptic curves over $\mathbb{Q}$