| L(s) = 1 | + (−0.587 + 0.261i)2-s + (0.365 − 1.91i)3-s + (−1.06 + 1.17i)4-s + (0.927 − 0.735i)5-s + (0.286 + 1.22i)6-s + (0.666 − 2.38i)7-s + (0.712 − 2.19i)8-s + (−0.757 − 0.299i)9-s + (−0.352 + 0.674i)10-s + (−0.351 − 0.122i)11-s + (1.87 + 2.46i)12-s + (−0.118 − 5.67i)13-s + (0.232 + 1.57i)14-s + (−1.07 − 2.04i)15-s + (−0.176 − 1.68i)16-s + (−1.30 + 1.33i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 + 0.184i)2-s + (0.211 − 1.10i)3-s + (−0.530 + 0.589i)4-s + (0.414 − 0.328i)5-s + (0.117 + 0.499i)6-s + (0.251 − 0.902i)7-s + (0.251 − 0.775i)8-s + (−0.252 − 0.0999i)9-s + (−0.111 + 0.213i)10-s + (−0.105 − 0.0368i)11-s + (0.540 + 0.712i)12-s + (−0.0329 − 1.57i)13-s + (0.0622 + 0.421i)14-s + (−0.276 − 0.529i)15-s + (−0.0441 − 0.420i)16-s + (−0.315 + 0.322i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.792616 - 0.491442i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.792616 - 0.491442i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-5.48 - 10.9i)T \) |
| good | 2 | \( 1 + (0.587 - 0.261i)T + (1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (-0.365 + 1.91i)T + (-2.78 - 1.10i)T^{2} \) |
| 5 | \( 1 + (-0.927 + 0.735i)T + (1.14 - 4.86i)T^{2} \) |
| 7 | \( 1 + (-0.666 + 2.38i)T + (-5.98 - 3.62i)T^{2} \) |
| 11 | \( 1 + (0.351 + 0.122i)T + (8.62 + 6.83i)T^{2} \) |
| 13 | \( 1 + (0.118 + 5.67i)T + (-12.9 + 0.544i)T^{2} \) |
| 17 | \( 1 + (1.30 - 1.33i)T + (-0.356 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.65 - 5.08i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.83 - 0.603i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (3.80 - 8.08i)T + (-18.4 - 22.3i)T^{2} \) |
| 31 | \( 1 + (-6.52 + 1.10i)T + (29.2 - 10.1i)T^{2} \) |
| 37 | \( 1 + (1.28 - 0.780i)T + (17.1 - 32.7i)T^{2} \) |
| 41 | \( 1 + (7.32 - 6.87i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (0.312 + 1.11i)T + (-36.7 + 22.2i)T^{2} \) |
| 47 | \( 1 + (0.954 + 0.288i)T + (39.1 + 26.0i)T^{2} \) |
| 53 | \( 1 + (8.93 + 1.12i)T + (51.3 + 13.1i)T^{2} \) |
| 59 | \( 1 + (-1.14 + 0.832i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-7.99 + 6.89i)T + (8.91 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-5.12 + 2.02i)T + (48.8 - 45.8i)T^{2} \) |
| 71 | \( 1 + (-5.72 - 5.85i)T + (-1.48 + 70.9i)T^{2} \) |
| 73 | \( 1 + (-2.78 + 0.715i)T + (63.9 - 35.1i)T^{2} \) |
| 79 | \( 1 + (-4.92 + 5.95i)T + (-14.8 - 77.6i)T^{2} \) |
| 83 | \( 1 + (0.0457 + 0.727i)T + (-82.3 + 10.4i)T^{2} \) |
| 89 | \( 1 + (-5.31 + 3.53i)T + (34.4 - 82.0i)T^{2} \) |
| 97 | \( 1 + (1.51 - 10.2i)T + (-92.8 - 28.0i)T^{2} \) |
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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
−13.11296123676364522393763162305, −12.22413331505188449797740329255, −10.63398480212073571467228064650, −9.660884315377420976727104903303, −8.246173640213013947249498313899, −7.79311588715523800759655323363, −6.76634093293841263012555623181, −5.12692604301332992326908384923, −3.43363553926501554859956105379, −1.21680981539951478821024466450,
2.31360549122345074673643153384, 4.33250392507709882349681475051, 5.22750399018193191561704577831, 6.64911190600334352543141434710, 8.565427602800655783827817522831, 9.318066390562277168547433881259, 9.854148121932543824475403121067, 10.95950100647511832280415217547, 11.81062928000076852030708553947, 13.51426388073855850136249106736
