| L(s) = 1 | + (−4.89 − 2.82i)2-s + (25.6 − 8.30i)3-s + (15.9 + 27.7i)4-s + (−9.39 + 5.42i)5-s + (−149. − 32.0i)6-s + (322. − 557. i)7-s − 181. i·8-s + (591. − 426. i)9-s + 61.3·10-s + (−1.05e3 − 611. i)11-s + (641. + 579. i)12-s + (1.16e3 + 2.01e3i)13-s + (−3.15e3 + 1.82e3i)14-s + (−196. + 217. i)15-s + (−512. + 886. i)16-s + 3.38e3i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.951 − 0.307i)3-s + (0.249 + 0.433i)4-s + (−0.0751 + 0.0433i)5-s + (−0.691 − 0.148i)6-s + (0.938 − 1.62i)7-s − 0.353i·8-s + (0.810 − 0.585i)9-s + 0.0613·10-s + (−0.795 − 0.459i)11-s + (0.371 + 0.335i)12-s + (0.529 + 0.917i)13-s + (−1.14 + 0.663i)14-s + (−0.0581 + 0.0643i)15-s + (−0.125 + 0.216i)16-s + 0.688i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.30134 - 0.816198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30134 - 0.816198i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 + 2.82i)T \) |
| 3 | \( 1 + (-25.6 + 8.30i)T \) |
| good | 5 | \( 1 + (9.39 - 5.42i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-322. + 557. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (1.05e3 + 611. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-1.16e3 - 2.01e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 3.38e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 6.23e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (1.24e4 - 7.18e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-1.17e4 - 6.78e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (4.39e3 + 7.60e3i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 2.78e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (4.79e4 - 2.76e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (2.69e4 - 4.66e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-1.52e5 - 8.80e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + 8.39e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-3.05e5 + 1.76e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (8.33e4 - 1.44e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-4.52e4 - 7.83e4i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 4.29e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.74e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (1.63e5 - 2.82e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (8.58e5 + 4.95e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + 5.38e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-4.29e4 + 7.44e4i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−17.39362139472225840713865577452, −15.94984681057312767040374797794, −14.19766451854885609312429074702, −13.35437430747913765655275502238, −11.37167682607086716879820466026, −10.04134438017136674673503354559, −8.301251528714429114827107632503, −7.30463858938699683081063075740, −3.82220532046763172225376006567, −1.42489293160621109759278876731,
2.36328460336228856892955483412, 5.29840352800090307521949783263, 7.86653001350429968364917711616, 8.755304089846000449083736004119, 10.24405208308614589571577438940, 12.09364499384252336866868448324, 13.99120802944588996407320190901, 15.34613871577669982270178831846, 15.80690421979813048849326903126, 18.05464821131698736934918990294
