I am looking for a slowly varying function $L(.)$ such that the function $f(s)=s+(1-s)^2L\left( \frac{1}{1-s}\right)$ is a probability generating function with $f'(1)=1$ and $f''(1)=\infty$.

I tryed $L(x)=\ln(x)$ but in this case $f(s)$ was not probability generating function.

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