Denote by $\zeta$ the Riemann zeta function.

It is known that

$$\int_{0}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2}).$$

But is a similar result for $\int_{0}^{T} \zeta(\sigma + it) \mathrm{d}t$, where $0<\sigma<1$ also known ?