| L(s) = 1 | + (15.4 − 11.2i)2-s + (−8.34 − 25.6i)3-s + (72.8 − 224. i)4-s + (0.205 + 0.149i)5-s + (−416. − 302. i)6-s + (242. − 746. i)7-s + (−635. − 1.95e3i)8-s + (−589. + 428. i)9-s + 4.84·10-s + (−4.41e3 + 157. i)11-s − 6.36e3·12-s + (3.85e3 − 2.80e3i)13-s + (−4.62e3 − 1.42e4i)14-s + (2.11 − 6.52i)15-s + (−7.30e3 − 5.31e3i)16-s + (8.29e3 + 6.02e3i)17-s + ⋯ |
| L(s) = 1 | + (1.36 − 0.990i)2-s + (−0.178 − 0.549i)3-s + (0.569 − 1.75i)4-s + (0.000735 + 0.000534i)5-s + (−0.787 − 0.572i)6-s + (0.267 − 0.822i)7-s + (−0.438 − 1.35i)8-s + (−0.269 + 0.195i)9-s + 0.00153·10-s + (−0.999 + 0.0356i)11-s − 1.06·12-s + (0.486 − 0.353i)13-s + (−0.450 − 1.38i)14-s + (0.000162 − 0.000499i)15-s + (−0.446 − 0.324i)16-s + (0.409 + 0.297i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.02827 - 3.10208i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02827 - 3.10208i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (8.34 + 25.6i)T \) |
| 11 | \( 1 + (4.41e3 - 157. i)T \) |
| good | 2 | \( 1 + (-15.4 + 11.2i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (-0.205 - 0.149i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-242. + 746. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-3.85e3 + 2.80e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-8.29e3 - 6.02e3i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-4.57e3 - 1.40e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 - 9.40e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-3.57e4 + 1.09e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-3.66e4 + 2.66e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (6.84e4 - 2.10e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (-1.65e5 - 5.10e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 5.79e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (1.84e4 + 5.66e4i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (1.37e6 - 9.98e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (2.93e5 - 9.02e5i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (2.12e6 + 1.54e6i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 + 3.39e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (4.89e4 + 3.55e4i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (5.68e5 - 1.74e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (3.05e6 - 2.21e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-4.99e6 - 3.63e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 1.26e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-8.22e6 + 5.97e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−14.14824892690820322103288064970, −13.31909868287685611337278163643, −12.45529503200801191888249288263, −11.16681790077072842182014275977, −10.31158685370779578344851957152, −7.82836984574518845583826744460, −5.99329317476009252969507448847, −4.56809806663286040455153551445, −2.90131766656803566810253646890, −1.13030989967285585652153489202,
3.11237291423369263613310950831, 4.85694074829894404836479139269, 5.71825722684802593479788250520, 7.32763350099410684822735915439, 8.956019567981833660443729935335, 10.99336178520253147919444902636, 12.33581200234815524940239203315, 13.44751070075977757367071300526, 14.65672383594157520902272065707, 15.53644817803427571232716065778
