Is there a way to tie up even and "newly suggested odd" Riemann zeta values? Define the sequence
$$a_s=(-1)^{\binom{s-1}2}\left(\frac{\pi}2\right)^s\frac1{2\cdot s!}\begin{cases} s\,E_{s-1}, \qquad \text{if $s$ is odd} \\ 2^{2s}B_s, \qquad \,\,\text{if $s$ is even};\end{cases}$$
where $E_s$ and $B_s$ are Euler and Bernoulli numbers, respectively.
Of course, we know that
$$\sum_{n=1}^{\infty}\frac1{n^{2s}}=a_{2s} \qquad \text{and} \qquad
\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{(2n-1)^{2s-1}}=a_{2s-1}.$$
The focus here is that each $a_s$ is rational multiple of a power of $\pi$ unlike the classical Riemann zeta function at odd values of $s$. A few examples are in order: for $s\geq1$,
$$\frac{\pi}4, \frac{\pi^2}6, \frac{\pi^3}{32}, \frac{\pi^4}{90}, \frac{5\pi^5}{1536}, \frac{\pi^6}{945}, \frac{61\pi^7}{184320}, \frac{\pi^8}{9450}, \,\,\dots$$
Now, I like to ask:

QUESTION. Is there a function $F(s)$, continuous in real $s$, whose values are $F(s)=a_s$ at integral $s$? One may try upgrading this to a meromorphic function.

 A: The function
$$f(s):=\sin(\tfrac{\pi}{2}(s+1)) \zeta(s)+\sin(\tfrac{\pi s}{2}) L(\tfrac{s+1}{2})\tag{1}$$
where $L(s)$ is the second Dirichlet series
$$L(s)=\sum\limits_{n=1}^{\infty } \frac{(-1)^{n-1}}{(2 n-1)^{2 s-1}}\tag{2}$$
which is valid for $\Re(s)>1$ but which can be analytically extended using the Hurwitz zeta function as
$$L(s)=\frac{\zeta \left(2 s-1,\frac{1}{4}\right)-\zeta \left(2 s-1,\frac{3}{4}\right)}{4^{2 s-1}}\tag{3}$$
gives the two sequences except for this sign
$$\underset{s\to 1}{\text{lim}}f(s)=-\frac{\pi }{4}, f(2)=\frac{\pi^2}{6}, f(3)=-\frac{\pi^3}{32}, f(4)=\frac{\pi^4}{90},\dots\tag{4}$$
and you can square the trigonometric functions to get the sign.
The plot of the function in Mathematica for 2<s<30 is a nice sine like wave, but this is all trivial.
Figure (1) below illustrates formula (1) for $f(s)$ in blue using formula (3) for $L(s)$ where the red discrete evaluation points illustrate the evaluation of formula (1) at integer values of $s$.


Figure (1): Illustration of Formula (1) for $f(s)$

Squaring the $sin$ terms in formula (1) for $f(s)$ above leads to $g(s)$ defined below which evaluates correctly in sign as well as magnitude.
$$g(s)=\sin^2\left(\frac{\pi}{2}(s+1)\right)\ \zeta(s)+\sin^2\left(\frac{\pi s}{2}\right)\ L\left(\frac{s+1}{2}\right)\tag{5}$$
Figure (2) below illustrates formula (5) for $g(s)$ in blue using formula (3) for $L(s)$ where the red discrete evaluation points illustrate the evaluation of $a_s$ defined in the question above at integer values of $s$.


Figure (2): Illustration of Formula (5) for $g(s)$ (blue curve) and $a_s$ (red evaluation points)

