Conjectural closed-form representations of sums, products or integrals

What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision anybody tried but still remain unproved?

I am also interested in values of special functions at certain points that have conjectural representations in terms of simpler functions (e.g. special values of hypergeometric functions, Meijer G-function or Fox H-function that representable in terms of elementary functions and well-known constants like $\pi$, $e$, Catalan, Euler–Mascheroni, Glaisher–Kinkelin or Khinchin).

To give some examples:

• Gourevitch conjecture mentioned at MathOverflow: $$\sum_{n=0}^\infty \frac{1+14n+76n^2+168n^3}{2^{20n}}\binom{2n}{n}^7\stackrel{?}{=}\frac{32}{\pi^3}.$$
• Riemann hypothesis (in an unusual form, also mentioned at MathOverflow) $$\int_{0}^{\infty}\frac{(1-12t^2)}{(1+4t^2)^3}\int_{1/2}^{\infty}\log|\zeta(\sigma+it)|~d\sigma ~dt\stackrel{?}{=}\frac{\pi(3-\gamma)}{32}.$$
• Another conjecture from MathOverflow attributed to J. M. Borwein, D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century: $$\frac{\displaystyle\int_{\pi/3}^{\pi/2}\log\left|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}\right|dt}{\displaystyle\sum_{n=1}^\infty\left(\frac n7\right)\frac{1}{n^2}}\stackrel{?}{=}\frac{7\sqrt{7}}{24},$$ where $(\frac{n}{7})$ denotes the Legendre symbol.

Some conjectural formulas for $\pi$ are given in:

• J. Guillera, "Kind of proofs of Ramanujan-like series"
• J. Guillera, "Mosaic supercongruences of Ramanujan-type"
• J. Guillera, "About a new kind of Ramanujan type series"
• J. Guillera, "Some challenging formulas for $\pi$"

For example (I expressed an infinite sum given in the paper in terms of hypergeometric functions): $$224 \, _5F_4\left(\frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{2}{3 };1,1,1,1;8235 \sqrt{5}-18414\right)\\-100 \sqrt{5} \, _5F_4\left(\frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{2}{3 };1,1,1,1;8235 \sqrt{5}-18414\right)\\-1655540 \, _5F_4\left(\frac{4}{3},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{5}{3 };1,2,2,2;\frac{27}{8} \left(5 \sqrt{5}-11\right)^3\right)\\+740380 \sqrt{5} \, _5F_4\left(\frac{4}{3},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{5}{3 };1,2,2,2;\frac{27}{8} \left(5 \sqrt{5}-11\right)^3\right)\\-1237563 \, _5F_4\left(\frac{4}{3},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{5}{3 };2,2,2,2;\frac{27}{8} \left(5 \sqrt{5}-11\right)^3\right)\\+553455 \sqrt{5} \, _5F_4\left(\frac{4}{3},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{5}{3 };2,2,2,2;\frac{27}{8} \left(5 \sqrt{5}-11\right)^3\right)\stackrel{?}{=}\frac{4}{\pi^2}$$

You can also look at:

• J. Borwein, D. Bradley, "Empirically determined Apéry-like formulae for ζ(4n + 3)"

Many such conjectures can be seen in Borwein-Crandall's recent text, Closed Forms: What They Are and Why We Care. One cited case is the identity $$F(3,5)= \frac{15}{\pi^2} \sum_{n=0}^\infty \binom{2n}{n}^2 \frac{(1/16)^{2n+1}}{2n+1},$$ in which $F(x,y)$ denotes the $4$-dimensional lattice sum: $$F(x,y)=(1+x)(1+y)\sum_{a,b,c,d} \frac{(-1)^{a+b+c+d}}{\left((6a+1)^2+x(6b+1)^2+y(6c+1)^2+xy(6d+1)^2\right)^2}.$$ Both expressions have been numerically estimated; each is approximately $0.0628326$.

A second example of historical interest is the well-known Wallis product: $$\frac{\pi}{2} =\frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5}\cdot \frac{6}{7}\cdot\frac{8}{7}\cdots \frac{2n}{2n-1}\cdot \frac{2n}{2n+1}\cdots$$ Wallis conjectured the given value $\pi/2$ after lengthy calculation in 1655. Methods did not exist to derive such a result at least until the days of Euler (about 80 years later).

Wallis' product is notable in its relation to Stirling's approximation (i.e. completing the transition between the weak and strong Stirling approximations), and for its role in determining $$\zeta'(0)=-\log(2\pi)/2.$$

You might want to have a look at the work of Zhi-Wei Sun and especially his 181 Conjectural Series for $\pi$ and Other Constants. Some examples:

$$\sum_{k=1}^\infty\frac{(10k-3)8^k}{k^3\binom{2k}{k}^2\binom{3k}{k}}=\frac{\pi^2}{2},$$

$$\sum_{k=1}^\infty\frac{(28k^2-18k+3)(-64)^k}{k^5\binom{2k}{k}^4\binom{3k}{k}}=-14\zeta(3),$$

$$\sum_{n=1}^\infty\frac{16n+5}{12^n}\sum_{k=1}^\infty\binom{2k}{k}\binom{2(n-k)}{n-k}\binom{-1/4}{k}\binom{-3/4}{n-k}=\frac{8}{\pi}.$$