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Here is one which I found at this MO link:

$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan(t)+\sqrt{7}}{\tan(t)-\sqrt{7}}\right|\ dt = \sum_{n\geq 1} \left(\frac n7\right)\frac{1}{n^2}, $$ where $\displaystyle\left(\frac n7\right)$ denotes the Legendre symbol. Not really my favorite identity, but it has the interesting feature that it is a conjecture! It is a rare example of a conjectured explicit identity between real numbers that can be checked to arbitrary accuracy. This identity has been verified to over 20,000 decimal places. See J. M. Borwein and D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, Natick, MA, 2004 (pages 90-91).

P.S. This problem was resolved before this post was placed in Section 5 of [D.H. Bailey, J.M. Borwein, D. Broadhurst and W. Zudilin, Experimental mathematics and mathematical physics, in "Gems in Experimental Mathematics", T. Amdeberhan, L.A. Medina, and V.H. Moll (eds.), Contemp. Math. 517 (2010), Amer. Math. Soc., 41–58]. In fact, the problem was solved even before its mentioning in the 2004 book; the details of the story can be found in the article.

Here is one which I found at this MO link:

$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan(t)+\sqrt{7}}{\tan(t)-\sqrt{7}}\right|\ dt = \sum_{n\geq 1} \left(\frac n7\right)\frac{1}{n^2}, $$ where $\displaystyle\left(\frac n7\right)$ denotes the Legendre symbol. Not really my favorite identity, but it has the interesting feature that it is a conjecture! It is a rare example of a conjectured explicit identity between real numbers that can be checked to arbitrary accuracy. This identity has been verified to over 20,000 decimal places. See J. M. Borwein and D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, Natick, MA, 2004 (pages 90-91).

P.S. This problem was resolved before this post was placed in Section 5 of [D.H. Bailey, J.M. Borwein, D. Broadhurst and W. Zudilin, Experimental mathematics and mathematical physics, in "Gems in Experimental Mathematics", T. Amdeberhan, L.A. Medina, and V.H. Moll (eds.), Contemp. Math. 517 (2010), Amer. Math. Soc., 41–58]. In fact, the problem was solved even before its mentioning in the 2004 book; the details of the story can be found in the article.

Here is one which I found at this MO link:

$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan(t)+\sqrt{7}}{\tan(t)-\sqrt{7}}\right|\ dt = \sum_{n\geq 1} \left(\frac n7\right)\frac{1}{n^2}, $$ where $\displaystyle\left(\frac n7\right)$ denotes the Legendre symbol. Not really my favorite identity, but it has the interesting feature that it is a conjecture! It is a rare example of a conjectured explicit identity between real numbers that can be checked to arbitrary accuracy. This identity has been verified to over 20,000 decimal places. See J. M. Borwein and D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, Natick, MA, 2004 (pages 90-91).

P.S. This problem was resolved before this post was placed in Section 5 of [D.H. Bailey, J.M. Borwein, D. Broadhurst and W. Zudilin, Experimental mathematics and mathematical physics, in "Gems in Experimental Mathematics", T. Amdeberhan, L.A. Medina, and V.H. Moll (eds.), Contemp. Math. 517 (2010), Amer. Math. Soc., 41–58]. In fact, the problem was solved even before its mentioning in the 2004 book; the details of the story can be found in the article.

Here is one which I found at this MO link:

$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan(t)+\sqrt{7}}{\tan(t)-\sqrt{7}}\right|\ dt = \sum_{n\geq 1} \left(\frac n7\right)\frac{1}{n^2}, $$ where $\displaystyle\left(\frac n7\right)$ denotes the Legendre symbol. Not really my favorite identity, but it has the interesting feature that it is a conjecture! It is a rare example of a conjectured explicit identity between real numbers that can be checked to arbitrary accuracy. This identity has been verified to over 20,000 decimal places. See J. M. Borwein and D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, Natick, MA, 2004 (pages 90-91).

P.S. This problem was resolved before this post was placed in Section 5 of [D.H. Bailey, J.M. Borwein, D. Broadhurst and W. Zudilin, Experimental mathematics and mathematical physics, in "Gems in Experimental Mathematics", T. Amdeberhan, L.A. Medina, and V.H. Moll (eds.), Contemp. Math. 517 (2010), Amer. Math. Soc., 41–58]. In fact, the problem was solved even before its mentioning in the 2004 book; the details of the story can be found in the article.

Here is one which I found at this MO link:

$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan(t)+\sqrt{7}}{\tan(t)-\sqrt{7}}\right|\ dt = \sum_{n\geq 1} \left(\frac n7\right)\frac{1}{n^2}, $$ where $\displaystyle\left(\frac n7\right)$ denotes the Legendre symbol. Not really my favorite identity, but it has the interesting feature that it is a conjecture! It is a rare example of a conjectured explicit identity between real numbers that can be checked to arbitrary accuracy. This identity has been verified to over 20,000 decimal places. See J. M. Borwein and D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, Natick, MA, 2004 (pages 90-91).

Here is one which I found at this MO link:

$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan(t)+\sqrt{7}}{\tan(t)-\sqrt{7}}\right|\ dt = \sum_{n\geq 1} \left(\frac n7\right)\frac{1}{n^2}, $$ where $\displaystyle\left(\frac n7\right)$ denotes the Legendre symbol. Not really my favorite identity, but it has the interesting feature that it is a conjecture! It is a rare example of a conjectured explicit identity between real numbers that can be checked to arbitrary accuracy. This identity has been verified to over 20,000 decimal places. See J. M. Borwein and D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, Natick, MA, 2004 (pages 90-91).

P.S. This problem was resolved before this post was placed in Section 5 of [D.H. Bailey, J.M. Borwein, D. Broadhurst and W. Zudilin, Experimental mathematics and mathematical physics, in "Gems in Experimental Mathematics", T. Amdeberhan, L.A. Medina, and V.H. Moll (eds.), Contemp. Math. 517 (2010), Amer. Math. Soc., 41–58]. In fact, the problem was solved even before its mentioning in the 2004 book; the details of the story can be found in the article.