modified riemann zeta function $\zeta ^*(s)$?

Question

I remember there being a function $\zeta ^*(s)$ where

$$\zeta ^*(s)=\zeta (s), \ s\neq 1$$

$$\zeta ^*(1)=\gamma$$

but now I can't seem to find any record of it, does a function like this exist or am I misremembering?

Answer 1

Defining $\zeta^*$ like that doesn't make much sense since the function won't even be continuous at $s=1$.

However, $\zeta^*(s) = \zeta(s) - \frac{1}{s-1}$ for $s\ne 1$ and $\zeta^*(1) = \gamma$ defines a function that is holomorphic at $s=1$.