| L(s) = 1 | + (2.62 + 8.06i)2-s + (7.28 + 5.29i)3-s + (−32.3 + 23.4i)4-s + (−8.26 + 25.4i)5-s + (−23.5 + 72.6i)6-s + (−58.4 + 42.4i)7-s + (−54.7 − 39.7i)8-s + (25.0 + 77.0i)9-s − 226.·10-s + (164. − 365. i)11-s − 359.·12-s + (4.13 + 12.7i)13-s + (−495. − 360. i)14-s + (−194. + 141. i)15-s + (−217. + 670. i)16-s + (167. − 515. i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 1.42i)2-s + (0.467 + 0.339i)3-s + (−1.01 + 0.734i)4-s + (−0.147 + 0.455i)5-s + (−0.267 + 0.823i)6-s + (−0.450 + 0.327i)7-s + (−0.302 − 0.219i)8-s + (0.103 + 0.317i)9-s − 0.717·10-s + (0.411 − 0.911i)11-s − 0.721·12-s + (0.00678 + 0.0208i)13-s + (−0.676 − 0.491i)14-s + (−0.223 + 0.162i)15-s + (−0.212 + 0.655i)16-s + (0.140 − 0.432i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.482895 + 2.01958i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.482895 + 2.01958i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-7.28 - 5.29i)T \) |
| 11 | \( 1 + (-164. + 365. i)T \) |
| good | 2 | \( 1 + (-2.62 - 8.06i)T + (-25.8 + 18.8i)T^{2} \) |
| 5 | \( 1 + (8.26 - 25.4i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (58.4 - 42.4i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-4.13 - 12.7i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-167. + 515. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-1.66e3 - 1.20e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 2.30e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (291. - 211. i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (1.79e3 + 5.53e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-5.59e3 + 4.06e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (2.06e3 + 1.49e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 7.99e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.09e4 + 7.97e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.20e4 + 3.71e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-4.02e4 + 2.92e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-5.07e3 + 1.56e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + 5.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-1.52e4 + 4.70e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (9.73e3 - 7.07e3i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-3.11e4 - 9.59e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (3.29e4 - 1.01e5i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 - 3.08e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-3.58 - 11.0i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−16.08500423784363273562241227278, −14.98221036522262811931328717718, −14.22594418511371780686609513980, −13.13141530607063234403692861789, −11.25046990831726968231393446359, −9.428963930714687374864741151275, −8.075134791416418925912192934025, −6.75938218675210059236932864812, −5.37751277106004330746937043989, −3.45607746784950988950583674750,
1.22969051733746093142222635700, 3.07129681407892862703316528046, 4.62230047190164038172622686266, 7.14143841503655274156539140287, 9.084115305651990912594967035157, 10.23365241959030366125242815229, 11.70840844113428515138390871484, 12.69036892209569733052312420713, 13.46795245754037531693898844249, 14.77329626169874693601413322185
