The mysterious expression, that Riemann did not "simplify enough to publish", is mentioned in his letter to Weierstrass published by Weber in 1876. It is reminiscent of Fermat's "marvelous proof" that the margin was "too narrow to contain", but a review of Riemann’s Nachlass led a librarian (Distel) to the likely culprit, apparently, unbeknownst to Hadamard. Siegel gave a reconstruction of it in On Riemann’s Nachlass for Analytic Number Theory (1932), and it is now known as the Riemann–Siegel formula:
"In fact, Herr Distel, a librarian, already several decades ago discovered the representation
in question of the zeta function in Riemann’s papers. It concerns an asymptotic development
which gives the behavior of the function $ζ(s)$ on the critical line $σ = 1/2$ and, more generally,
in each strip $σ_1 ≤ σ ≤ σ_2$, expressed for infinitely increasing magnitude of $s$. The principal
term of this development was meanwhile rediscovered, independently of Riemann, by Hardy
and Littlewood in 1920 as a consequence of their “approximate functional equation”; they use
the same methods of proof as Riemann, namely approximate calculation of an integral by the
saddle-point method. In Riemann, however, there is also a process for obtaining additional
terms of the asymptotic series, which is based on the beautiful properties of the integral
$$Φ(τ, u)=\int\frac{e^{\pi i\tau x^2+2\pi iux}}{e^{2\pi ix}-1}dx,$$
which, incidentally, has also led Kronecker and recently Mordell to a most elegant derivation
of the reciprocity formula for the Gauss sums.
In 1926 Bessel-Hagen noted in a new review of
the Riemann papers another previously unknown representation of the zeta function in terms of definite integrals; in this Riemann was also
guided by the properties of $Φ(τ, u)$.
These two representations of $ζ(s)$ may be included among the most important results in
Riemann’s number-theoretic Nachlass, insofar as one does not find them in his published paper."
The first "two or three" properties that Riemann derived from these expressions are likely that the Riemann xi function $\xi(s)$ is an entire function of $s^2$, with a Weierstrassian product expansion, which Hadamard rigorously proved in 1893, and an asymptotic form of the prime number theorem that he derived from it in 1896, see Iurato, On some historical aspects of the theory of Riemann zeta function, pp. 77-78, 104. Both are traceable to Riemann's 1859 paper, and Iurato specifically alludes to Hadamard's remark from The Psychology of Invention in the Mathematical Field (1945) in connection with them:
"Following (Stopple 2003, Chapter 10, Section 10.1), it was Riemann to
realize that a product formula for ξ(s) would have had a great significance for the study of prime numbers. The first rigorous proof of this product formula was due to Hadamard but, as he himself remembered, it took almost three decades before he reached to a satisfactory proof of it. Likewise, also H.M. Edwards (1974, Chapter 1, Sections 1.8-1.19) affirms that the parts concerned with (2) are the most difficult portion of the 1859 Riemann's paper (see also (Bottazzini & Gray 2013, Chapter 5, Section
5.10)). Their goal is to prove essentially that $ξ(s)$ can be expressed as an infinite product... "Hadamard (in 1893) proved necessary and sufficient conditions for the validity of the product formula $ξ(t) = ξ(0) ∏_ρ(1 − t/ρ)$ but the steps of the argument
by which Riemann went from the one to the other are obscure, to say the very least"...
Just in regard to the novelties due to Hadamard, Weyl points out that the driving force for these Hadamard's investigations was the wish to obtain sufficient information about the zeros of the Riemann zeta function for establishing the asymptotic law for the distribution of prime numbers... In 1896, both Hadamard and
de la Vallée-Poussin, independently of each other, were able to draw the conclusion $\left[\frac{\pi(n)\ln(n)}{n}\to1\right]$ from 1893 Hadamard's results concerning entire functions".
