I am reading Serre's article on p-adic Modular forms (Formes modulaires et fonctions zˆeta p-adiques).
In section 1.6, Serre constructs p-adic Eisenstein series by considering
$G_{k_i} = \frac{1}{2}\zeta(1-k_i) + \sum_{n \geq 1} \sigma_{k_i-1}(n) q^n$
such that $k_i$ converge to $k$ in $X = \mathbb{Z}_p \times \mathbb{Z}/(p-1)\mathbb{Z}$ with $|k_i|_{\infty}$ tending to infinity.
This sequence of q-expansions converge p-adically and thus we arrive at a p-adic Eisenstein series. Hence, for a fixed $k \in X$ we have been able to define a p-adic Eisenstein series.
Following this, Serre provides one specific example.
Let $p \equiv 3 (mod.4)$ and let $k = (1,\frac{p+1}{2}) \in X$, then by previous procedure we get a p-adic Eisenstein series which is
$\frac{1}{2}h(-p) + \sum_{n \geq 1} \sum_{d|n} (\frac{d}{p}) q^n$
where $h(-p)$ denotes the class number of $\mathbb{Q}(\sqrt{-p})$.
I think I can figure out the coefficients of $q^n$ for $n \geq 1$ but I have no clue about the constant term. How do we get this?
And I feel this is due to my lack of knowledge about "some relation between special values of zeta and class numbers".
Let $\lambda: (\mathbb{Z}/p)^{\times} \rightarrow \mathbb{C}^{\times}$ be the unique character of order two.
Consider, for $s \in \mathbb{C}$, the series $E_s(\tau)=\sum_{p \mid c}{\frac{y^{s-1}\lambda(d)}{(c\tau+d)|c\tau+d|^{2(s-1)}}}$, for $\tau \in \mathbb{H}$. It is absolutely convergent for $Re(s) >>0$, smooth as a function of $\tau$ and holomorphic with respect to $s$.
Its Fourier expansion can be explicitly computed for $Re(s)>>0$ and meromorphically continued to $\mathbb{C}$ with some explicit bounds. Since $E_s$, while not holomorphic, transforms under $\Gamma_0(p)$ like an element of $\mathcal{M}_1(\Gamma_1(p),\lambda)$, we see that $E_s$ has meromorphic continuation to $s \in \mathbb{C}$, with an explicit Fourier expansion, and it still transforms in the same way.
It can be checked in particular with the analytic class number formula (probably in eg Neukirch’s Algebraic Number Theory) that $E_1(\tau)$ is proportional to $E(\tau) := \frac{h(-p)}{2} +\sum_{n \geq 1}{\sum_{d \mid n}{\lambda(d)}q^n}$, hence $E(\tau) \in \mathcal{M}_1(\Gamma_0(p),\lambda)$.
(The computation above should be carried out in more detail in most texts about classical modular forms, eg Diamond-Shurman or Miyake, making the connection to the analytic class number formula more apparent).
In particular, $E^2 \in \mathcal{M}_2(\Gamma_0(p))$, so it is a $p$-adic modular form in the sense of Serre.
Let $F$ be the $p$-adic limit of the $G_{1+(p-1)p^n/2}$ (it exists, by Corollaire 2, p. 204). It’s not too hard to see that in $\mathbb{Z}_p[[q]]$, $E-F$ is a constant $\alpha \in \mathbb{Z}_p$ (see 4), p. 208). But of course, $E$ has no $p$-adic meaning (while $F$ is purely $p$-adic).
However, $E^2,F^2$ are $p$-adic modular forms of weight $2$, and $E^2-F^2=\alpha (E+F)=\alpha (2F+\alpha)$. If $\alpha \neq 0$, it follows that $2F+\alpha$ is a $p$-adic modular form of weight $2$.
Reducing mod $p$, the contradiction follows from the previous results of Swinnerton-Dyer (see the bottom of p. 196).