| L(s) = 1 | + (−17.1 − 12.4i)2-s + (8.34 − 25.6i)3-s + (100. + 308. i)4-s + (190. − 138. i)5-s + (−464. + 337. i)6-s + (−503. − 1.54e3i)7-s + (1.28e3 − 3.95e3i)8-s + (−589. − 428. i)9-s − 5.01e3·10-s + (4.39e3 + 361. i)11-s + 8.74e3·12-s + (−5.60e3 − 4.06e3i)13-s + (−1.06e4 + 3.29e4i)14-s + (−1.96e3 − 6.05e3i)15-s + (−3.80e4 + 2.76e4i)16-s + (1.72e4 − 1.25e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.52 − 1.10i)2-s + (0.178 − 0.549i)3-s + (0.781 + 2.40i)4-s + (0.682 − 0.495i)5-s + (−0.877 + 0.637i)6-s + (−0.554 − 1.70i)7-s + (0.888 − 2.73i)8-s + (−0.269 − 0.195i)9-s − 1.58·10-s + (0.996 + 0.0819i)11-s + 1.46·12-s + (−0.707 − 0.513i)13-s + (−1.04 + 3.20i)14-s + (−0.150 − 0.463i)15-s + (−2.32 + 1.68i)16-s + (0.849 − 0.617i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.224901 + 0.629159i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.224901 + 0.629159i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (-4.39e3 - 361. i)T \) |
| good | 2 | \( 1 + (17.1 + 12.4i)T + (39.5 + 121. i)T^{2} \) |
| 5 | \( 1 + (-190. + 138. i)T + (2.41e4 - 7.43e4i)T^{2} \) |
| 7 | \( 1 + (503. + 1.54e3i)T + (-6.66e5 + 4.84e5i)T^{2} \) |
| 13 | \( 1 + (5.60e3 + 4.06e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-1.72e4 + 1.25e4i)T + (1.26e8 - 3.90e8i)T^{2} \) |
| 19 | \( 1 + (6.11e3 - 1.88e4i)T + (-7.23e8 - 5.25e8i)T^{2} \) |
| 23 | \( 1 + 7.21e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-2.47e3 - 7.61e3i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-2.73e4 - 1.98e4i)T + (8.50e9 + 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-1.26e4 - 3.89e4i)T + (-7.68e10 + 5.57e10i)T^{2} \) |
| 41 | \( 1 + (6.77e4 - 2.08e5i)T + (-1.57e11 - 1.14e11i)T^{2} \) |
| 43 | \( 1 + 9.41e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.54e5 + 7.83e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-4.57e5 - 3.32e5i)T + (3.63e11 + 1.11e12i)T^{2} \) |
| 59 | \( 1 + (2.24e5 + 6.89e5i)T + (-2.01e12 + 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-2.49e6 + 1.81e6i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 - 1.97e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-2.32e6 + 1.69e6i)T + (2.81e12 - 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-3.53e5 - 1.08e6i)T + (-8.93e12 + 6.49e12i)T^{2} \) |
| 79 | \( 1 + (1.77e6 + 1.28e6i)T + (5.93e12 + 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-5.28e5 + 3.84e5i)T + (8.38e12 - 2.58e13i)T^{2} \) |
| 89 | \( 1 - 4.60e4T + 4.42e13T^{2} \) |
| 97 | \( 1 + (2.59e6 + 1.88e6i)T + (2.49e13 + 7.68e13i)T^{2} \) |
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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
−13.97299074810350513301517800104, −12.85228749230150040251645396136, −11.78531806315643494734126337997, −10.16729899611088811081837254675, −9.649203153278079239924444308572, −8.060233045694848200455381732735, −6.93207606491304923537025867630, −3.55115073495216277816300075082, −1.61271213348580698621090632876, −0.50155519949755585602433137470,
2.16696441975270558391836554566, 5.67889135207688206063436895377, 6.57625854341225715476659080387, 8.440025141076301970322627515237, 9.412893700904499458025037678166, 10.08363950818677101212247432549, 11.83030266710742570354040904300, 14.31971772455952614540418993146, 15.02608023438175144509861651627, 16.03193262505063914975858243509
