Yes, you can do a lot of what you ask about in your question, and if you're willing to change the questions slightly, you can do even more. I have thought a lot about these kinds of questions. I'm sorry this will be a long answer in which I cite my own papers a lot.
A few years ago I thought specifically about the kind of idea you describe, using complex-analytic methods to handcraft an "ad hoc" zeta-function to have desired special values which would recover orders of homotopy groups of my favorite spectra. You get a theory of "gamma-functions of spectra," so-called because in the simplest case you recover the classical gamma-function--see below. I got far enough to give a talk about it, and I found my notes from that talk, so I'm posting them below, at the end of this answer.
However, in the end I didn't like the theory of "gamma-functions of spectra," so I never wrote it up. The problem is that the gamma-functions aren't well-behaved enough to do anything surprising with them. Instead I tried other approaches, and found one that I think works much better, building zeta-functions and L-functions of spectra using algebraic methods rather than purely analytic methods. To be clear, by "algebraic methods," I mean that instead of using complex analysis to craft an "ad hoc" meromorphic function with specified special values, I go looking for some algebraic gadget, like an Iwasawa module or a number field or a Galois representation or a family of Dirichlet characters, with an associated zeta- or L-function whose special values in the left half-plane I can calculate, and I try to find some conceptual way to match the algebraic gadget to the spectrum, so that the special values of the L-function have DENOMINATORS which agree with the orders of the (appropriately Bousfield-localized) homotopy groups of some spectrum whose homotopy groups I can also calculate. This has disadvantages because it's usually pretty nontrivial to figure out how to make that "matching" work, and even when it works, you don't have a topological interpretation of the NUMERATORS of the special values of the L-function!
Nevertheless I feel the results I've found by this algebraic method have been much better than what I was able to find by the "ad hoc" complex-analytic approach. Those "better results" are in:
my papers https://arxiv.org/abs/2303.09550 and https://arxiv.org/abs/2308.01805 , both published, which set up a theory of "$KU$-local zeta-functions" of finite spectra, with lots of good properties: analytic continuation, nice Euler products, functional equations, special values have denominators that recover the orders of the $KU$-local homotopy groups of the spectrum, useful applications to number theory (e.g. proving some cases of the Leopoldt conjecture as a consequence of periodicity in $K(1)$-local stable homotopy groups)
also in the preprint https://arxiv.org/abs/2311.10225 , part of which is joint with Matthias Strauch, which shows that this theory of zeta-functions of spectra is compatible with a certain kind of "topological Jacquet-Langlands duality" (i.e., there's an automorphic rep of $GL_1$ associated to the $\hat{\mathbb{G}}_m$-based TJL dual of a finite spectrum, and the Euler factor at each prime is equal to the corresponding Euler factor of the $KU$-local zeta-function of that finite spectrum),
and also in the preprint https://arxiv.org/abs/2408.02163 , joint with my student Austin Maison, in which we set up some of the $p$-adic theory, building a very simple theory of algebraic $p$-adic $L$-functions of spectra using Iwasawa theory, with a lot of the same good properties as the complex "$KU$-local" zeta-functions of spectra built in the papers I already mentioned.
I think that all paints a reasonable picture of some kind of theory of zeta-functions for finite spectra. But the theory described is all height 1, either $KU$-local (in the complex-analytic setting where we want to capture all primes) or $K(1)$-local (in the $p$-adic analytic setting where we want to focus on a single prime). I know how to make some of this story work at higher heights, but I don't know how to make it look nice! At primes $p>3$, you can match up the orders of the $K(2)$-local homotopy groups of the mod $p$ Moore spectrum with factors in special values of various Dirichlet characters, but you need several Dirichlet characters to do it--it's as though different parts of $\pi_*(L_{K(2)}S/p)$ come from special values of different $L$-functions. I plan to post something about that this year. Maison and I are also using Venjakob's noncommutative Iwasawa theory to try to do something in the direction of $K(2)$-local $p$-adic $L$-functions. So this story isn't nearly done yet.
Anyway, below are my notes on "gamma-functions of spectra," which, as I said, I felt were kind of a dead end. I haven't thought about that stuff in a few years--I hope there's nothing truly stupid in there!
There is a "cheap" analytic way to construct a meromorphic function on the complex plane whose values at integers recover the orders of homotopy groups of Bousfield-localized finite spectra. These are the gamma-functions of spectra, of which the classical $\Gamma$-function is a special case. There are some interesting features of these gamma-functions, but they do not seem to connect as compellingly to problems in number theory or reveal new phenomena in stable homotopy groups as much as the algebraically-constructed zeta-functions of spectra.
Here's a new definition:
let $X$ be a spectrum, and let $\zeta\in\mathbb{C}$ be a root of unity.
The $\zeta$-twisted gamma-function of $X$ is defined as
$\Gamma(s,X,\zeta) = \int_0^{\infty} x^{s-1}\left(\sum_{j\geq 0} \frac{(\zeta x)^j}{j!\ \#(\pi_{j}(X))}\right)\ dx.$
Clearly $\Gamma(s,X,\zeta)$ only makes sense when $\pi_{j}(X)$ is finite for all $j\geq 0$.
The function $\Gamma(s,X,\zeta)$ is a function of the complex variable $s$, but there's no guarantee it converges anywhere at all---for example, if $\zeta$ is an $n$th root of unity, then for any prime $p$, $\Gamma(s,\coprod_{k\geq 0} \Sigma^{nk} H\mathbb{F}_p, \zeta)$ diverges for all $s$. If $\zeta=1$, it's even worse: $\Gamma(s,pt.,1)$ diverges for all $s$ (since it's the Mellin transform of $e^x$).
Ramanujan's "Master Theorem" (Ramanujan-Hardy): let $M_f(s) = \int_0^{\infty} x^{s-1}\left(\sum_{k\geq 0} \frac{(-1)^kf(k)}{k!}\right)\ dx$. Then $M_f(s) = \Gamma(s)f(-s)$ wherever $\Gamma(s+1)f(s)$ satisfies a bound on growth in vertical strips in the complex plane.
So $\#\pi_{j}(X) = (-1)^j \zeta^j\Gamma(-j)/\Gamma(-j,X,\zeta)$ whenever the right-hand side makes sense (i.e., whenever $\Gamma(s,X,\zeta)$ analytically continues into the complex numbers with negative real part and the singularity in the RHS at nonnegative integers $j$ is removable).
It is a special property of the "monochromatic layers" of finite spectra that, unlike arbitrary spectra, they actually have exactly the right properties for that RHS to make sense:
Theorem (S., 2017): If $p > 1+\frac{n^2+n}{2}$ and $\zeta$ is a primitive $4(p^n-1)$st root of unity and $X$ is the $E(n)$-localization of an $E(n-1)$-acyclic finite complex, then $\Gamma(s,X,\zeta)$ analytically continues to a meromorphic function on the whole complex plane, satisfying:
$\#\pi_{j}(X) = (-1)^j \zeta^j\Gamma(-j)/\Gamma(-j,X,\zeta)$ for all integers $j$ (including negative integers!),
$\Gamma(s+j,\Sigma^{k}X, \zeta) = (-1)^j\zeta^j \Gamma(s,X,\zeta) \prod_{k=1}^j (s+k)$ for all positive integers $j$,
There exists some positive integer $N$ such that $\Gamma(s+N,X,\zeta) = \Gamma(s,X,\zeta)\prod_{k=0}^{N-1}(s+k)$.
Notice that the third condition,
in the case $N=1$, is exactly the usual functional equation for the classical gamma-function: $\Gamma(s+1) = s\Gamma(s)$. In fact $\Gamma(s,pt.,-1) = \Gamma(s)$, so the gamma-functions of spectra indeed are a generalization of the classical gamma-function.
Notice that the two conditions
$\#\pi_{j}(X) = (-1)^j \zeta^j\Gamma(-j)/\Gamma(-j,X,\zeta)$ for all integers $j$ (including negative integers!),
There exists some positive integer $N$ such that $\Gamma(s+N,X,\zeta) = \Gamma(s,X,\zeta)\prod_{k=0}^{N-1}(s+k)$.
...together imply some kind of periodicity in the homotopy groups of $X$. In fact this periodicity is precisely the periodicity given by Ravenel's periodicity conjecture: since $X$ is $E(n-1)$-acyclic, it has type $\geq n$, so either it is $E(n)$-locally contractible or admits a $v_n$-power self-map.
That's actually how you prove the 2017 theorem from several slides ago:the periodicity conjecture/theorem gives you a $v_n$-power self-map on $X$, so it induces an iso of $K(n)_*K(n)$ comodules from $K(n)_*(X)$ to $\Sigma^{N}K(n)_*(X)$. The $E(n-1)$-acyclicity of $X$, together with an interlude on a Hilbert-Kunz variant of local Hochschild cohomology, gets you from there to having a iso of $E(n)_*E(n)$-comodules from $E(n)_*(X)$ to $\Sigma^{M}E(n)_*(X)$, for some multiple $M$ of $N$. Now the low vanishing line in the $E(n)$-Adams spectral sequence for $X$ is enough to give you the equality $\#(\pi_*(L_{E(n)}X)) = \#(\pi_*(L_{E(n)}\Sigma^MX))$, and consequently that $\sum_{j\geq 0} \frac{(\zeta x)^j}{j!\ \#(\pi_{j}(X))}$ satisfies the differential equation $y^{(M)}+y = 0$. By functional properties of the Mellin transform, that ODE then gives you the functional equation $\Gamma(s+N,X,\zeta) = \Gamma(s,X,\zeta)\prod_{k=0}^{N-1}(s+k)$ and consequently the analytic continuation you need in order to use Ramanujan's master theorem.
Interesting (very fun proof!), but probably not that powerful a result, since the really big insights in analytic number theory come from the kinds of manipulations you can make with functions that are Dirichlet series with known analytic continuations and Euler products.
We saw that the gamma-functions of spectra have good analytic continuations, and their special values do indeed count orders of Bousfield-localized stable homotopy groups of finite spectra (which is what we wanted), but these gamma-functions almost certainly are not equal to Dirichlet series, and almost certainly do not admit Euler products.
If we build zeta-functions of spectra algebraically (instead of these "gamma-functions of spectra"), we get Dirichlet series, Euler products, and analytic continuation. The price we pay is that we have to do some work and bring in some new ideas make those algebraic constructions, and when we do it, we lose control over the NUMERATORS of the special values: the denominators recover the orders of Bousfield-localized homotopy groups of finite spectra, but the numerators wind up being enormous products of irregular primes for which I have no topological interpretation.
Sorry again that this answer is so long and so full of self-citation!
