Let $f(x)=\frac{x}{e^x-1}=\sum_{k=0}^\infty \frac{(-1)^k B_k}{k!}(-x)^k$. It is well-kown that $$\int_0^\infty x^{s-2}f(x)dx=\Gamma(s)\zeta(s).\tag 1$$ On the other hand by Ramanujan's Master Theorem, we have $$\int_0^\infty x^{s-2}f(x)dx=\Gamma(s-1)B_{1-s}.\tag2$$ From $(1)$ and $(2)$, we deduce that $$\zeta(s)=\frac{(-1)^{1-s}B_{1-s}}{s-1}.$$ Now, $B_{1-s}$ is an entire function and it is a famous conjecture that its non-trivial zeros lie on the line $\Re s=\frac12.$
My question: Is there something like $B_s$, $s\in\Bbb C$ in the literature? I can't think of anything more than Bernoulli numbers!
Thanks for any help.