Background

Rational zeta series are series of the form $$\sum_{n=2}^{\infty} q_{n} \zeta(n + p, m), \label{1} \tag{1} $$ where $\zeta(x,m)$ is the Hurwitz zeta function and $q_{n}, \ p \in \mathbb{Q} \setminus \{ 0 \} $ for all $n$.

Every real number $x$ can be expressed by means of such a series. For instance, we have $$ \sum_{n=2}^{\infty} \zeta(n,2) = \sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) =1. \tag{2}\label{2} $$ Here, $\zeta(\cdot)$ is the Riemann zeta function. Also, we have $$ \sum_{n=2}^{\infty} \frac{(\zeta(n)-1)}{n} = 1 - \gamma, \tag{3}\label{3} $$ where $\gamma$ is the Euler-Mascheroni constant.

Zeta values in mathematics

Geometrical and probabilistic interpretations of zeta values at positive integer arguments can be found in this MO question. The appear in Tamagawa number calculuations, quantum gauge theories, distributions of Cauchy random variables, probabilities involving $n$-free numbers, and more.

Mathematical interpretations of the negative Riemann zeta values can be found in this MO question. They appear, for instance, in the orbifold characteristic of genus g Riemann surfaces.

Rational zeta series at non-integer arguments

The paragraphs above provide examples of zeta values that come up in mathematics. In the context of this question, I am not interested in such isolated values, but in the rational zeta series described at the top. More specifically, I am interested in rational zeta series with non-integer arguments. What I mean by that, is that $p \in \mathbb{Q} \setminus \mathbb{Z}$ in equation \eqref{1}.

Examples of such series come up in the study of the Spiral of Theodorus. For instance, the Theodorus constant is described in equation (16) of following paper (PDF) by David Brink:

\begin{align} T &:= \frac{1}{2} + \sum_{k=1}^{\infty} (-1)^{k+1} \left[ \zeta\left(k+ \frac{1}{2} \right) -1 \right] \tag{4} \label{4} \\ &= \sum_{x=1}^{\infty} \frac{1}{(x+1)\sqrt{x}}. \tag{5} \label{5} \end{align}

It is related to the sum of the internal angles of Spiral of Theodorus, depicted below:

$\hspace{4cm}$

Another example of a rational zeta series with non-integer arguments is Hlawka's Schneckenconstante (described on p. 2 of Brink's paper):

\begin{align} K &:= \sum_{k=0}^{\infty} (-1)^{k} \frac{\zeta(k+1/2)}{2k+1} \tag{6}\label{6} \\ &= \frac{\pi}{4} + \sum_{k=0}^{\infty} (-1)^{k} \frac{\zeta(k+1/2) -1}{2k+1}. \tag{7}\label{7} \end{align}

Questions

1. What other rational zeta series at non-integer arguments (see equation \eqref{1} with $p \in \mathbb{Q} \setminus \mathbb{Z}$) come up as mathematical constants? How and where do such series appear?
2. And what about the case where $p \in \mathbb{R} \setminus \mathbb{Q} $ ?

Note: I've asked a similar question at Physics SE.