In this paper J.B. Keiper defined the following function:
$$\tau_k = \sum_{j=1}^k (-1)^j\,{k-1 \choose j-1} \sigma_{j+1} \qquad k \ge 1, k \in \mathbb{N} \tag{1}$$
where $\displaystyle\sigma_r = \sum_{\rho}\frac{1}{\rho^r}$, with $\rho$ a non-trivial zero of $\zeta(s)$ and taken in pairs $\rho, 1- \rho$.
Keiper also notes that $\tau_k = \lambda_{k+1} + \lambda_{k-1} -2\lambda_k$ where $\lambda_k$ is Li's constant ($=k\lambda_k^{Keiper})$. These $\lambda_k$ can be expressed in terms of the (paired) non-trivial zeros:
$$\lambda_k = \sum_{\rho,1-\rho} \left(1-\left(\frac{\rho}{\rho-1}\right)^k+1-\left(\frac{\rho-1}{\rho}\right)^k \right)$$
and from this it follows that $\lambda_{-k} = \lambda_{k}$ and $\tau_{-k} = \tau_{k}$.
Numerical evidence suggests this could be continued further into this entire function:
$$\tau_s = \sum_{j=1}^\infty\, (-1)^j\, {s-1 \choose j-1} \sigma_{j+1} \qquad s \in \mathbb{C} \tag{2} $$
Below is a plot to illustrate its behaviour. The RH implies the oscillations must stay (at integer values) within the two bounds and vice versa:
With $\tau_s$ being an entire function, it should have a Hadamard product factorisation into its zeros. After computing the first few of them, I found only a single complex zero $z=55.309... + 12.826...i$ (i.e. inducing a party of 4 complex zeros in total). All other zeros appear to be real and come in pairs $(\mu_{n},\mu_{-n})$, with $\mu_{1..24}$ shown below:
All the typical $e^{(.)}$-factors in the Hadamard product nicely cancel out and what remains is:
$$\tau_s = \tau_0\left(1-\frac{s^2}{z^2}\right)\left(1-\frac{s^2}{\overline{z^2}}\right)\prod_n\left(1-\frac{s^2}{\mu_n^2}\right)\left(1-\frac{s^2}{\overline{\mu_n^2}}\right) \tag{3}$$
where $\tau_0 = \sigma_1 = 1 + \dfrac{\gamma-\log(\pi)}{2}-\log(2)$. Numerical results look pretty good already at 24 real zeros.
Q: Is there anything more to learn from the existence of such a canonical Hadamard product? Could it say anything about the oscillating behaviour of $\tau_s$ or the distribution of its zeros?
Note 1:
$\tau_s$ could also be expressed in terms of $\rho$'s:
$$\tau_s = \sum_{\rho,1-\rho} \left(\frac{\left(\frac{\rho}{\rho-1}\right)^s}{\rho^2}+\frac{\left(\frac{\rho-1}{\rho}\right)^s}{(1-\rho)^2)}\right)\qquad s\in \mathbb{C}, \Re(s) \ge 0$$
$$\tau_s = \sum_{\rho,1-\rho} \left(\frac{\left(\frac{\rho}{\rho-1}\right)^s}{(1-\rho)^2}+\frac{\left(\frac{\rho-1}{\rho}\right)^s}{\rho^2}\right)\qquad s\in \mathbb{C}, \Re(s) \le 0 $$
Note 2:
Also found that $\tau_s$ could be split into a smooth first term and an oscillating second term:
$$\tau_s=\sum_{k=1}^{\infty} \left( {s+k-1\choose k-1}\cdot \frac{\zeta(k+1)}{(-2)^{k+1}}- {-s+k-1\choose k-1} \sigma_{k+1}^*\right)$$
with $\sigma_k^*$ defined here. The first term is asymptotic to $\dfrac{1-2^{-s}}{2s}$ for $\Re(s) \gt 0$.
Note 3:
It seems also possible to expand the domain of $\lambda_k$:
$$\lambda_s = \sum_{j=1}^{\infty} \, (-1)^{j+1}\, {s \choose j} \sigma_{j} \qquad s \in \mathbb{C} \tag{4}$$
and of $\lambda_k - \lambda_{k+1}$:
$$\lambda_s - \lambda_{s+1} = \sum_{j=1}^\infty\,{s + j - 1 \choose s} \sigma_{j} \qquad s \in \mathbb{C} \tag{5} $$
However, their Hadamard products turn out to be less elegantly compared to the one for $\tau_s$. Product (4) seems to have only complex zeros (which makes sense, since the positivity of $\lambda_k$ for all $k$ also implies the RH, hence no real zeros are expected to be found). Product (5) also appears to have a single real zero at $s=-\frac12$ and complex zeros otherwise.
