Derivation of the Dirichlet L-series of order $1$

Question

I would like to know how the Dirichlet L-series(of order $1$) were derived. I independently found sequences analogous to the Dirichlet L-series using a property from: https://math.stackexchange.com/q/4776903

Basically it was proved that $k-\sin(x)=k\cdot(1-\dfrac{1}{x_{root1}})(1-\dfrac{1}{x_{root2}})(1-\dfrac{1}{x_{root3}})\cdots$, For $|k|<1$

For instance, $\dfrac{1}{\sqrt{2}}-\sin(x)=\dfrac{1}{\sqrt{2}}\cdot\left(1-\dfrac{4x}{\pi}\right)\left(1-\dfrac{4x}{3\pi}\right)\left(1+\dfrac{4x}{5\pi}\right)\left(1+\dfrac{4x}{7\pi}\right) \cdots$

By taking the logarithm of both the sides, we get:

$\log{(\dfrac{1}{\sqrt{2}}-\sin(x))}=\log{(\dfrac{1}{\sqrt{2}})}+\log\left(1-\dfrac{4x}{\pi}\right)+\log\left(1-\dfrac{4x}{3\pi}\right)+\log\left(1+\dfrac{4x}{5\pi}\right)+\log\left(1+\dfrac{4x}{7\pi}\right) \cdots$

Differentiating and substituting $x=0$ reveals:

$\dfrac{\pi}{2\sqrt{2}}=1+\dfrac{1}{3}-\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{9}+\cdots$

This is apparently $L_{-8}(1)$ according to https://mathworld.wolfram.com/DirichletL-Series.html (31st formula). One could replace $\dfrac{-1}{\sqrt{2}}$ with any constant $k,(|k|<1)$ to derive other Dirichlet L-series of order $1$. I would like to know if this is indeed how the Dirichlet L-series were derived and if my derivation is novel.

I have searched online for the derivation, however I couldn't find any references to the derivation of the series.

I would greatly appreciate any derivations of the Dirichlet L-series of order $1$ :)

Answer 1

If $d$ is a negative fundamental discriminant, then $$L(1,\chi_d)=\frac{\pi}{(2-\chi_d(2))\sqrt{-d}}\sum_{t=1}^{\lfloor -d/2\rfloor}\chi_d(t).$$ Here $\chi_d$ denotes the quadratic Dirichlet character $\left(\frac{d}{\cdot}\right)$. For a proof, see the proof of Theorem 3.9 in Section 10.3 of Rose: A course in number theory (2nd ed.).

Example. For $d=-8$ the above formula gives that $$L(1,\chi_{-8})=\frac{\pi}{2\sqrt{8}}(1+0+1+0)=\frac{\pi}{2\sqrt{2}}.$$