| L(s) = 1 | + (0.326 + 1.37i)2-s + (−1.18 + 2.75i)3-s + (−1.78 + 0.897i)4-s + (−3.75 + 2.79i)5-s + (−4.17 − 0.735i)6-s + (1.03 − 0.679i)7-s + (−1.81 − 2.16i)8-s + (−6.18 − 6.54i)9-s + (−5.07 − 4.26i)10-s + (−0.479 + 4.10i)11-s + (−0.351 − 5.98i)12-s + (−1.23 − 4.12i)13-s + (1.27 + 1.20i)14-s + (−3.24 − 13.6i)15-s + (2.38 − 3.20i)16-s + (−7.03 − 1.24i)17-s + ⋯ |
| L(s) = 1 | + (0.163 + 0.688i)2-s + (−0.395 + 0.918i)3-s + (−0.446 + 0.224i)4-s + (−0.751 + 0.559i)5-s + (−0.696 − 0.122i)6-s + (0.147 − 0.0971i)7-s + (−0.227 − 0.270i)8-s + (−0.686 − 0.726i)9-s + (−0.507 − 0.426i)10-s + (−0.0436 + 0.373i)11-s + (−0.0292 − 0.499i)12-s + (−0.0950 − 0.317i)13-s + (0.0909 + 0.0857i)14-s + (−0.216 − 0.911i)15-s + (0.149 − 0.200i)16-s + (−0.414 − 0.0730i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 + 0.613i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.202553 - 0.590534i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.202553 - 0.590534i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.326 - 1.37i)T \) |
| 3 | \( 1 + (1.18 - 2.75i)T \) |
| good | 5 | \( 1 + (3.75 - 2.79i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (-1.03 + 0.679i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (0.479 - 4.10i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (1.23 + 4.12i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (7.03 + 1.24i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (1.18 + 6.73i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (18.9 - 28.8i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (8.94 - 8.43i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (0.301 - 5.16i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (13.6 - 4.98i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (17.2 - 72.6i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (15.2 + 35.3i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-5.69 + 0.331i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (-69.6 - 40.2i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-6.72 - 57.5i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (-2.15 - 1.08i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (-80.9 + 85.7i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (-3.83 + 4.56i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (73.3 - 61.5i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-71.4 + 16.9i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (25.0 + 105. i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (-16.7 - 19.9i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-22.3 + 30.0i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−13.40289669962308828687837573797, −12.04097798432595943323717945245, −11.25651531691406789584790687922, −10.21952415454348107853853007191, −9.171596491837364725726773923372, −7.930267552960560679669837771523, −6.89799054356866924384563878891, −5.61076093415613673677296525005, −4.45325092498610733408004785475, −3.35647543028664019252449917996,
0.37993220247405788032882113091, 2.15022842218353781080498431944, 4.01897963890798918863078977792, 5.31835396907437968445908925785, 6.62873935332851936383065726119, 8.032966553650317206644706293810, 8.717356946013546745590230451003, 10.30484778432653847767720845004, 11.38250776614451568945638602645, 12.04322683645886178394370373714
