| L(s) = 1 | + (−7.93 + 5.76i)2-s + (2.78 + 8.55i)3-s + (19.8 − 61.0i)4-s + (−37.2 − 27.0i)5-s + (−71.4 − 51.8i)6-s + (37.2 − 114. i)7-s + (97.6 + 300. i)8-s + (−65.5 + 47.6i)9-s + 451.·10-s + (319. + 242. i)11-s + 578.·12-s + (488. − 354. i)13-s + (365. + 1.12e3i)14-s + (128. − 394. i)15-s + (−845. − 614. i)16-s + (−1.12e3 − 817. i)17-s + ⋯ |
| L(s) = 1 | + (−1.40 + 1.01i)2-s + (0.178 + 0.549i)3-s + (0.620 − 1.90i)4-s + (−0.666 − 0.484i)5-s + (−0.809 − 0.588i)6-s + (0.287 − 0.884i)7-s + (0.539 + 1.66i)8-s + (−0.269 + 0.195i)9-s + 1.42·10-s + (0.795 + 0.605i)11-s + 1.15·12-s + (0.801 − 0.582i)13-s + (0.498 + 1.53i)14-s + (0.147 − 0.452i)15-s + (−0.826 − 0.600i)16-s + (−0.944 − 0.686i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0456i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.662895 - 0.0151383i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.662895 - 0.0151383i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.78 - 8.55i)T \) |
| 11 | \( 1 + (-319. - 242. i)T \) |
| good | 2 | \( 1 + (7.93 - 5.76i)T + (9.88 - 30.4i)T^{2} \) |
| 5 | \( 1 + (37.2 + 27.0i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-37.2 + 114. i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-488. + 354. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (1.12e3 + 817. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (258. + 794. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 - 3.26e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-2.61e3 + 8.04e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-5.17e3 + 3.76e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (3.63e3 - 1.11e4i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (3.29e3 + 1.01e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 2.64e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (4.45e3 + 1.37e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (3.20e4 - 2.32e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-3.12e3 + 9.61e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-2.12e4 - 1.54e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + 1.06e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-6.33e3 - 4.60e3i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.35e4 + 4.15e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (5.28e4 - 3.84e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (1.35e4 + 9.81e3i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 - 5.53e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (2.34e4 - 1.70e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−15.71850531908973184117274361252, −15.24111466540558429232542852696, −13.65780412729542088205849844424, −11.43896005210422613694885541542, −10.19487180236731243134753599583, −8.972896491513527342574577586722, −7.977354590354724074640228307465, −6.67528932816705848316726403527, −4.49221091101285667318668370421, −0.68979503476020814811808867932,
1.51146708257951135248596827811, 3.27778754906266652264062745880, 6.74787029873784952244195016926, 8.389612327763237939572069538856, 8.994574626193827419881459168534, 10.92305319791342251159422601824, 11.55236815269070896121634973738, 12.69635979380368804050026922639, 14.52765967393908474214167765937, 16.02435704992927326209699628051
