| L(s) = 1 | + (13.8 + 7.93i)2-s + (−47.1 − 113. i)3-s + (130. + 220. i)4-s + (−368. + 889. i)5-s + (248. − 1.95e3i)6-s + (1.42e3 − 1.42e3i)7-s + (57.9 + 4.09e3i)8-s + (−6.11e3 + 6.11e3i)9-s + (−1.21e4 + 9.43e3i)10-s + (−2.21e3 + 5.34e3i)11-s + (1.89e4 − 2.52e4i)12-s + (1.99e4 + 4.82e4i)13-s + (3.11e4 − 8.51e3i)14-s + 1.18e5·15-s + (−3.16e4 + 5.73e4i)16-s + 9.93e4i·17-s + ⋯ |
| L(s) = 1 | + (0.868 + 0.495i)2-s + (−0.582 − 1.40i)3-s + (0.508 + 0.861i)4-s + (−0.589 + 1.42i)5-s + (0.191 − 1.51i)6-s + (0.595 − 0.595i)7-s + (0.0141 + 0.999i)8-s + (−0.931 + 0.931i)9-s + (−1.21 + 0.943i)10-s + (−0.151 + 0.365i)11-s + (0.915 − 1.21i)12-s + (0.699 + 1.68i)13-s + (0.812 − 0.221i)14-s + 2.34·15-s + (−0.483 + 0.875i)16-s + 1.18i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0418 - 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.0418 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.46211 + 1.40212i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46211 + 1.40212i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-13.8 - 7.93i)T \) |
| good | 3 | \( 1 + (47.1 + 113. i)T + (-4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (368. - 889. i)T + (-2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (-1.42e3 + 1.42e3i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 + (2.21e3 - 5.34e3i)T + (-1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (-1.99e4 - 4.82e4i)T + (-5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 - 9.93e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (-5.31e4 + 2.20e4i)T + (1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (8.66e3 + 8.66e3i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 + (3.60e5 - 1.49e5i)T + (3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 + 1.15e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-9.11e5 + 2.19e6i)T + (-2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (-4.77e5 + 4.77e5i)T - 7.98e12iT^{2} \) |
| 43 | \( 1 + (1.60e6 - 3.86e6i)T + (-8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 + 8.82e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (-1.19e7 - 4.93e6i)T + (4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (8.20e5 + 3.39e5i)T + (1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (-1.90e7 + 7.89e6i)T + (1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (-7.37e6 - 1.78e7i)T + (-2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (6.11e6 - 6.11e6i)T - 6.45e14iT^{2} \) |
| 73 | \( 1 + (-3.08e7 + 3.08e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 - 8.75e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + (-3.81e7 + 1.58e7i)T + (1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (2.48e7 + 2.48e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 + 3.83e7T + 7.83e15T^{2} \) |
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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
−14.91890033745533874607685227068, −14.07966613632583042390596901910, −13.00255223333157750502661224284, −11.58528115859243006347426618701, −11.11308982539044428675958306938, −7.87056114731653180378852571778, −7.03727708834029407495802092703, −6.21106310360501347181085830577, −3.98749681888615679860595132201, −1.95443511992508751990634387735,
0.71715724018200300532478582069, 3.50023575117209556320993528512, 5.00520185779100748576482756263, 5.38802303513311988839633441398, 8.450569202808053772159538331621, 9.893387736295269814342608742009, 11.19393613277187203554814589081, 12.01979619728208362008476870035, 13.29283024322117816861862479765, 15.05752980905800656314577993361
