| 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
−14.77329626169874693601413322185, −13.46795245754037531693898844249, −12.69036892209569733052312420713, −11.70840844113428515138390871484, −10.23365241959030366125242815229, −9.084115305651990912594967035157, −7.14143841503655274156539140287, −4.62230047190164038172622686266, −3.07129681407892862703316528046, −1.22969051733746093142222635700,
3.45607746784950988950583674750, 5.37751277106004330746937043989, 6.75938218675210059236932864812, 8.075134791416418925912192934025, 9.428963930714687374864741151275, 11.25046990831726968231393446359, 13.13141530607063234403692861789, 14.22594418511371780686609513980, 14.98221036522262811931328717718, 16.08500423784363273562241227278
