The Ramanujan Summation of some infinite sums is consistent with a couple of sets of values of the Riemann zeta function. We have, for instance, $\zeta(-2n)=\sum_{n=1}^{\infty} n^{2k} = 0 (\mathfrak{R}) $ (for non-negative integer $k$) and $\zeta(-(2n+1))=-\frac{B_{2k}}{2k} (\mathfrak{R})$ (again, $k \in \mathbb{N} $). Here, $B_k$ is the $k$'th Bernoulli number. However, it does not hold when, for example, $\sum_{n=1}^{\infty} \frac{1}{n}=\gamma (\mathfrak{R})$ (here $\gamma$ denotes the Euler-Mascheroni Constant) as it is not equal to $\zeta(1)=\infty$.
Question: Are the first two examples I stated the only instances in which the Ramanujan summation of some infinite series coincides with the values of the Riemann zeta function?