| L(s) = 1 | + (−4.12 − 7.13i)2-s + (−23.0 − 13.3i)3-s + (−1.97 + 3.41i)4-s + (68.5 − 39.5i)5-s + 219. i·6-s + (337. + 62.3i)7-s − 495.·8-s + (−9.46 − 16.3i)9-s + (−565. − 326. i)10-s + (854. − 1.47e3i)11-s + (90.9 − 52.5i)12-s + 3.12e3i·13-s + (−945. − 2.66e3i)14-s − 2.10e3·15-s + (2.16e3 + 3.75e3i)16-s + (3.52e3 + 2.03e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.515 − 0.892i)2-s + (−0.854 − 0.493i)3-s + (−0.0307 + 0.0533i)4-s + (0.548 − 0.316i)5-s + 1.01i·6-s + (0.983 + 0.181i)7-s − 0.966·8-s + (−0.0129 − 0.0224i)9-s + (−0.565 − 0.326i)10-s + (0.641 − 1.11i)11-s + (0.0526 − 0.0303i)12-s + 1.42i·13-s + (−0.344 − 0.971i)14-s − 0.625·15-s + (0.528 + 0.916i)16-s + (0.718 + 0.414i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.392441 - 0.725960i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.392441 - 0.725960i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-337. - 62.3i)T \) |
| good | 2 | \( 1 + (4.12 + 7.13i)T + (-32 + 55.4i)T^{2} \) |
| 3 | \( 1 + (23.0 + 13.3i)T + (364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (-68.5 + 39.5i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-854. + 1.47e3i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 - 3.12e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-3.52e3 - 2.03e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-5.08e3 + 2.93e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (6.66e3 + 1.15e4i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 - 6.51e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + (1.03e4 + 5.99e3i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-2.32e3 - 4.01e3i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 - 1.93e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 9.16e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (5.58e4 - 3.22e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (7.47e4 - 1.29e5i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-5.28e4 - 3.05e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (8.54e4 - 4.93e4i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.55e5 + 2.70e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 4.01e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (5.82e5 + 3.36e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-1.60e5 - 2.77e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 8.32e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-3.28e5 + 1.89e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 1.05e6iT - 8.32e11T^{2} \) |
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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
−20.78965722205587640760447996730, −19.08522690273589804101437867361, −18.00902535343896506852554774437, −16.79197978979911381534904226460, −14.27772228077999752197697306604, −12.02469954154241117430666997732, −11.16143097698837600049576952302, −9.092525462748972678962028411747, −5.99406924309418701021070553956, −1.30502508342190098794537976476,
5.57324835393045564923158776619, 7.68496695202889220788515278751, 10.04192242648353687159038956289, 11.86703896165371847302246270065, 14.51524115741466337099841943885, 15.97358525768217344279719003086, 17.49623702039371620406802091964, 17.79189939412151491478449576421, 20.49725066381162632822203702931, 21.92492369889387006153809046895
