| L(s) = 1 | + (2.14 − 2.95i)2-s + (−17.7 + 5.76i)3-s + (5.76 + 17.7i)4-s + (52.2 + 19.8i)5-s + (−21.0 + 64.8i)6-s + 140. i·7-s + (176. + 57.1i)8-s + (84.8 − 61.6i)9-s + (171. − 111. i)10-s + (−518. − 376. i)11-s + (−204. − 281. i)12-s + (600. + 826. i)13-s + (415. + 302. i)14-s + (−1.04e3 − 51.7i)15-s + (64.6 − 46.9i)16-s + (−392. − 127. i)17-s + ⋯ |
| L(s) = 1 | + (0.379 − 0.522i)2-s + (−1.13 + 0.369i)3-s + (0.180 + 0.554i)4-s + (0.934 + 0.355i)5-s + (−0.238 + 0.735i)6-s + 1.08i·7-s + (0.972 + 0.315i)8-s + (0.349 − 0.253i)9-s + (0.540 − 0.353i)10-s + (−1.29 − 0.938i)11-s + (−0.409 − 0.563i)12-s + (0.985 + 1.35i)13-s + (0.566 + 0.411i)14-s + (−1.19 − 0.0594i)15-s + (0.0631 − 0.0458i)16-s + (−0.329 − 0.107i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.22927 + 0.697027i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22927 + 0.697027i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-52.2 - 19.8i)T \) |
| good | 2 | \( 1 + (-2.14 + 2.95i)T + (-9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (17.7 - 5.76i)T + (196. - 142. i)T^{2} \) |
| 7 | \( 1 - 140. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (518. + 376. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-600. - 826. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (392. + 127. i)T + (1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-338. + 1.04e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-2.26e3 + 3.12e3i)T + (-1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-479. - 1.47e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (22.2 - 68.5i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (421. + 580. i)T + (-2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.07e4 + 7.80e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 6.36e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (1.01e4 - 3.28e3i)T + (1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-4.30e3 + 1.39e3i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-1.93e4 + 1.40e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (4.13e4 + 3.00e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-3.55e3 - 1.15e3i)T + (1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-1.20e4 - 3.70e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-1.40e4 + 1.93e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (732. + 2.25e3i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-2.42e4 - 7.87e3i)T + (3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-6.00e3 - 4.36e3i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-2.29e4 + 7.47e3i)T + (6.94e9 - 5.04e9i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.65317191118280931999192482916, −15.88636741578032668004657507115, −13.88875917470550146234416318431, −12.76464661261048019674339690003, −11.33809204858781738557789933472, −10.79098953538704340320922582806, −8.794658501276199692587013535874, −6.37855565588784492295681390061, −5.06005089107032194969363188415, −2.59823333169250302783901305611,
1.06746587166675451430577251344, 5.10000900583581076655162137439, 6.01283852016525969563061950181, 7.44290155507566059230506577369, 10.13035313410657610181704405673, 10.85672262928119587946942070116, 12.90148044208335946050891735096, 13.56604420852093820854984424122, 15.22733552488908655328313148690, 16.42428294346769946828275965448
