any integer argument is given by E. A. Karatsuba.

Negative integers

In general, for negative integers, one has

The so-called "trivial zeros" occur at the negative even integers: (Ramanujan summation)

The first few values for negative odd integers are

This means can be used as the definition of Bernoulli numbers.

However, just like the Bernoulli numbers, these do not stay small for increasingly negative odd