Graph of price of CDS against par spread

Question

I'm new to credit and I'm trying to wrap my head around the following idea. I understand that the par spread $s$ is the value of the fixed coupon payment at which the fixed and floating legs are equal in present value. Therefore, for a given CDS with coupon $c$, I should expect a price of $c-s$. Therefore, if I plot the price against par spread, I should see a linear graph with $y$-intercept $c$ and $x$-intercept $s$; however, I know there behaviour eventually changes due to the recovery $R$. How do I relate $(1-R)$ with $s$? I know $s$ and $r$ are related via $s(t,T)=\frac{V^{\textrm{floating}}(t,T)}{V^{\textrm{fixed}}(t,T)}=\frac{(1-R)\mathbb{E}^\mathbb{Q}\left[P(t,\tau)\mathbb{1}_{\tau\leq T_n}\Big|\mathcal{F}_t\right]}{\mathbb{E}^\mathbb{Q}\left[\sum_{i=1}^n(T_i-T_{i-1})P(t,T_i)\mathbb{1}_{\tau>T_i}\Big|\mathcal{F}_t\right]}$ but I'm pretty sure it should be simpler than this and I'm overthinking. Any help is appreciated, thanks!