A rational Ramanujan-like series for $\pi^{-m}$ and character $\chi$ is a series with rational parameters which is of the following form:

$$ \sum_{n=0}^{\infty} \left(\prod_{i=0}^{2m} \frac{(s_i )_{n}}{(1)_{n}} \right) \sum_{k=0}^{m} a_k n^k z_0^{n} = \frac{\sqrt{(-1)^m \chi}}{\pi^m}.$$

Consider the following extension with $x$:

$$ \sum_{n=0}^{\infty} \left(\prod_{i=0}^{2m} \frac{(s_i )_{n+x}}{(1)_{n+x}} \right) \sum_{k=0}^{m} a_k (n+x)^k z_0^{n+x}, $$

and replace $z_0^{n+x}$ with $(-1)^n (-z_0)^x$ if $z_0<0$. I can justify by means of a $p$-adic argument that for any particular series the coefficient of $x^{2m+1}$ in the development of the series is of the form $r L(\chi, m+1)$, where $r$ is a rational number, but I do not know how to prove that this is the case for all these kind of series.

We show two examples:

1-) If we develop in powers of $x$ the series

$$ \sum_{n=0}^{\infty} \frac{\left(\frac12\right)_{n+x}\left(\frac14\right)_{n+x}\left(\frac34\right)_{n+x}}{(1)_{n+x}^3}(65(n+x)+8) (-1)^n \left( \frac{16^2}{63^2} \right)^{n+x}, $$

that has $\chi=-7$, then the coefficient of $x^3$ is a rational multiple of $L(-7, 2)$.

2-) If we develop in powers of $x$ the series:

\begin{multline} \sum_{n=0}^{\infty} \frac{\left( \frac12 \right)_{n+x} \left( \frac13 \right)_{n+x} \left( \frac23 \right)_{n+x} \left( \frac16 \right)_{n+x} \left(\frac56 \right)_{n+x}}{(1)_{n+x}^5} (-1)^n \left( \frac{1}{80^3} \right)^{n+x} \\ \left(5418(n+x)^2+693(n+x)+29\right), \end{multline}

that has $\chi=5$, then the coefficient of $x^5$ is a rational multiple of $L(5, 3)$.

See arXiv:2312.16827

Any ideas?