| L(s) = 1 | + (−4.89 + 2.82i)2-s + (−22.0 + 12.7i)3-s + (15.9 − 27.7i)4-s + (−85.1 − 147. i)5-s + (71.8 − 124. i)6-s + 473.·7-s + 181. i·8-s + (−41.7 + 72.2i)9-s + (834. + 481. i)10-s − 2.11e3·11-s + 813. i·12-s + (2.40e3 + 1.38e3i)13-s + (−2.32e3 + 1.33e3i)14-s + (3.74e3 + 2.16e3i)15-s + (−512. − 886. i)16-s + (4.19e3 + 7.27e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.814 + 0.470i)3-s + (0.249 − 0.433i)4-s + (−0.681 − 1.18i)5-s + (0.332 − 0.576i)6-s + 1.38·7-s + 0.353i·8-s + (−0.0572 + 0.0990i)9-s + (0.834 + 0.481i)10-s − 1.59·11-s + 0.470i·12-s + (1.09 + 0.630i)13-s + (−0.845 + 0.488i)14-s + (1.11 + 0.641i)15-s + (−0.125 − 0.216i)16-s + (0.854 + 1.48i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.730129 + 0.426099i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.730129 + 0.426099i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 - 2.82i)T \) |
| 19 | \( 1 + (-6.81e3 + 734. i)T \) |
| good | 3 | \( 1 + (22.0 - 12.7i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (85.1 + 147. i)T + (-7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 - 473.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 2.11e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (-2.40e3 - 1.38e3i)T + (2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + (-4.19e3 - 7.27e3i)T + (-1.20e7 + 2.09e7i)T^{2} \) |
| 23 | \( 1 + (-3.49e3 + 6.05e3i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.54e4 - 1.46e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 - 3.85e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 4.91e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-3.95e4 + 2.28e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-3.40e4 - 5.89e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (7.90e4 - 1.36e5i)T + (-5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-6.07e4 - 3.50e4i)T + (1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-2.14e5 + 1.23e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-2.59e4 + 4.48e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.44e5 + 8.35e4i)T + (4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + (-2.28e3 + 1.31e3i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (-1.15e5 - 2.00e5i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (3.76e5 - 2.17e5i)T + (1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 5.53e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + (-3.24e5 - 1.87e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (-2.74e5 + 1.58e5i)T + (4.16e11 - 7.21e11i)T^{2} \) |
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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
−15.92638464055995813316608105270, −14.28519029224205988828526948233, −12.56528682292648931585205057340, −11.27614284098599791538358791050, −10.51645645370924673977614471506, −8.538418317772267794818371357096, −7.926155010862608229090760676487, −5.54949815919079860340862773016, −4.66256463042599209159683033881, −1.14054043821287817513721788011,
0.72831394409718254240022842354, 2.99156869507346526541003749994, 5.45417340733362768623970168291, 7.30336202409764549770771444019, 8.066354026030000649610056668357, 10.28171718655111099806704918775, 11.35960902086193018599076392507, 11.72171104670191722027987741114, 13.52920435157042736469323955710, 15.02339312775822790163985365257
