| L(s) = 1 | + (7.27 − 8.66i)2-s + (13.3 − 79.8i)3-s + (−22.2 − 126. i)4-s + (258. − 711. i)5-s + (−595. − 696. i)6-s + (690. − 3.91e3i)7-s + (−1.25e3 − 724. i)8-s + (−6.20e3 − 2.12e3i)9-s + (−4.28e3 − 7.41e3i)10-s + (9.36e3 + 2.57e4i)11-s + (−1.03e4 + 98.2i)12-s + (3.45e4 − 2.90e4i)13-s + (−2.89e4 − 3.44e4i)14-s + (−5.34e4 − 3.01e4i)15-s + (−1.53e4 + 5.60e3i)16-s + (6.12e4 − 3.53e4i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (0.164 − 0.986i)3-s + (−0.0868 − 0.492i)4-s + (0.414 − 1.13i)5-s + (−0.459 − 0.537i)6-s + (0.287 − 1.63i)7-s + (−0.306 − 0.176i)8-s + (−0.946 − 0.324i)9-s + (−0.428 − 0.741i)10-s + (0.639 + 1.75i)11-s + (−0.499 + 0.00473i)12-s + (1.21 − 1.01i)13-s + (−0.753 − 0.897i)14-s + (−1.05 − 0.595i)15-s + (−0.234 + 0.0855i)16-s + (0.733 − 0.423i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0297i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.999 - 0.0297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0427853 + 2.87596i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0427853 + 2.87596i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.27 + 8.66i)T \) |
| 3 | \( 1 + (-13.3 + 79.8i)T \) |
| good | 5 | \( 1 + (-258. + 711. i)T + (-2.99e5 - 2.51e5i)T^{2} \) |
| 7 | \( 1 + (-690. + 3.91e3i)T + (-5.41e6 - 1.97e6i)T^{2} \) |
| 11 | \( 1 + (-9.36e3 - 2.57e4i)T + (-1.64e8 + 1.37e8i)T^{2} \) |
| 13 | \( 1 + (-3.45e4 + 2.90e4i)T + (1.41e8 - 8.03e8i)T^{2} \) |
| 17 | \( 1 + (-6.12e4 + 3.53e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (6.79e4 - 1.17e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-1.76e5 + 3.10e4i)T + (7.35e10 - 2.67e10i)T^{2} \) |
| 29 | \( 1 + (5.77e5 - 6.88e5i)T + (-8.68e10 - 4.92e11i)T^{2} \) |
| 31 | \( 1 + (-3.66e4 - 2.07e5i)T + (-8.01e11 + 2.91e11i)T^{2} \) |
| 37 | \( 1 + (-6.81e5 - 1.18e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (2.76e5 + 3.30e5i)T + (-1.38e12 + 7.86e12i)T^{2} \) |
| 43 | \( 1 + (1.34e6 - 4.89e5i)T + (8.95e12 - 7.51e12i)T^{2} \) |
| 47 | \( 1 + (-4.35e6 - 7.68e5i)T + (2.23e13 + 8.14e12i)T^{2} \) |
| 53 | \( 1 + 4.62e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-2.57e6 + 7.07e6i)T + (-1.12e14 - 9.43e13i)T^{2} \) |
| 61 | \( 1 + (-4.16e4 + 2.36e5i)T + (-1.80e14 - 6.55e13i)T^{2} \) |
| 67 | \( 1 + (-1.91e7 + 1.60e7i)T + (7.05e13 - 3.99e14i)T^{2} \) |
| 71 | \( 1 + (-3.71e7 + 2.14e7i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (9.53e6 - 1.65e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-2.53e7 - 2.12e7i)T + (2.63e14 + 1.49e15i)T^{2} \) |
| 83 | \( 1 + (4.49e7 - 5.35e7i)T + (-3.91e14 - 2.21e15i)T^{2} \) |
| 89 | \( 1 + (-4.16e7 - 2.40e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (8.58e7 - 3.12e7i)T + (6.00e15 - 5.03e15i)T^{2} \) |
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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
−12.93726684131396316141496444337, −12.38342603769200019341270594650, −10.85899314911054724427007971285, −9.598493801347303789457466090013, −8.095853775550494426571685623140, −6.84051588143840199351203569342, −5.18914629074992648330219166831, −3.77006389383881436947125961238, −1.54835647217790519136127455683, −0.957292136891916134373912572726,
2.59945405410264058090911195694, 3.75399423217055683249051995156, 5.66471230609176311416989816907, 6.28229438871135467071267719029, 8.505775907881877060456757622076, 9.172730291572501689104624644138, 11.00884157098562948592571241199, 11.57664799288023082511738212406, 13.57229141477462694363171923569, 14.44206535102591722411654111248
