There are two similar functions; they determine the dependence of the values ​​of similar equations

For positive parameters, in the region of convergence of their power series, both have two positive roots.

Through power series we can only obtain zero roots of these functions

We need to find an analytical method, a formula for obtaining the first root of zero. It is known that the dependence of these roots is the same for both functions

I tried this: let's take special cases s=1 L=.01 and L=.1

a=0.1118325591589629648 b=3.5771520639572972184 b/a=31.9866780377671590 A=0.01010152719853875327 B=6.47277512439400469474 B/A=640.7719344982143479 It is very similar that in the formula that shows the dependence of the first root on the zero root, the natural logarithm of 1/L divided by L is used 10ln10=23.02585092994 100ln100=460.5170185988091368 There are still some factors and/or terms missing