| L(s) = 1 | + (−2.44 − 3.36i)2-s + (1.60 + 4.94i)3-s + (−0.408 + 1.25i)4-s + (−26.0 − 18.8i)5-s + (12.7 − 17.4i)6-s + (−15.8 − 5.13i)7-s + (−58.1 + 18.8i)8-s + (−21.8 + 15.8i)9-s + 133. i·10-s + (−113. − 40.8i)11-s − 6.86·12-s + (−8.90 − 12.2i)13-s + (21.3 + 65.8i)14-s + (51.6 − 158. i)15-s + (222. + 161. i)16-s + (314. − 432. i)17-s + ⋯ |
| L(s) = 1 | + (−0.611 − 0.841i)2-s + (0.178 + 0.549i)3-s + (−0.0255 + 0.0785i)4-s + (−1.04 − 0.755i)5-s + (0.353 − 0.485i)6-s + (−0.322 − 0.104i)7-s + (−0.907 + 0.294i)8-s + (−0.269 + 0.195i)9-s + 1.33i·10-s + (−0.941 − 0.337i)11-s − 0.0476·12-s + (−0.0526 − 0.0725i)13-s + (0.109 + 0.335i)14-s + (0.229 − 0.706i)15-s + (0.870 + 0.632i)16-s + (1.08 − 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0394i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.00963208 - 0.487758i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00963208 - 0.487758i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.60 - 4.94i)T \) |
| 11 | \( 1 + (113. + 40.8i)T \) |
| good | 2 | \( 1 + (2.44 + 3.36i)T + (-4.94 + 15.2i)T^{2} \) |
| 5 | \( 1 + (26.0 + 18.8i)T + (193. + 594. i)T^{2} \) |
| 7 | \( 1 + (15.8 + 5.13i)T + (1.94e3 + 1.41e3i)T^{2} \) |
| 13 | \( 1 + (8.90 + 12.2i)T + (-8.82e3 + 2.71e4i)T^{2} \) |
| 17 | \( 1 + (-314. + 432. i)T + (-2.58e4 - 7.94e4i)T^{2} \) |
| 19 | \( 1 + (-243. + 79.1i)T + (1.05e5 - 7.66e4i)T^{2} \) |
| 23 | \( 1 + 385.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-902. - 293. i)T + (5.72e5 + 4.15e5i)T^{2} \) |
| 31 | \( 1 + (849. - 616. i)T + (2.85e5 - 8.78e5i)T^{2} \) |
| 37 | \( 1 + (-576. + 1.77e3i)T + (-1.51e6 - 1.10e6i)T^{2} \) |
| 41 | \( 1 + (696. - 226. i)T + (2.28e6 - 1.66e6i)T^{2} \) |
| 43 | \( 1 + 1.29e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (30.6 + 94.1i)T + (-3.94e6 + 2.86e6i)T^{2} \) |
| 53 | \( 1 + (4.21e3 - 3.06e3i)T + (2.43e6 - 7.50e6i)T^{2} \) |
| 59 | \( 1 + (-390. + 1.20e3i)T + (-9.80e6 - 7.12e6i)T^{2} \) |
| 61 | \( 1 + (-2.44e3 + 3.37e3i)T + (-4.27e6 - 1.31e7i)T^{2} \) |
| 67 | \( 1 + 3.68e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-4.67e3 - 3.39e3i)T + (7.85e6 + 2.41e7i)T^{2} \) |
| 73 | \( 1 + (8.35e3 + 2.71e3i)T + (2.29e7 + 1.66e7i)T^{2} \) |
| 79 | \( 1 + (-2.85e3 - 3.93e3i)T + (-1.20e7 + 3.70e7i)T^{2} \) |
| 83 | \( 1 + (-3.63e3 + 4.99e3i)T + (-1.46e7 - 4.51e7i)T^{2} \) |
| 89 | \( 1 - 3.27e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-5.14e3 + 3.73e3i)T + (2.73e7 - 8.41e7i)T^{2} \) |
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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
−15.80156210335999255822722845992, −14.20643047293968584081838084960, −12.45157224142371081099932388258, −11.48646740854701054626288330527, −10.24392881551703723534278769835, −9.127120154657915538084148893616, −7.83697875582145700485207053083, −5.20118018255815416706877695513, −3.18103597065940857125446589372, −0.40899983571423503300637140433,
3.28320874631173429946688269176, 6.21909644895732390960283944319, 7.58532255464083871453507077557, 8.143093371883556473983156745988, 10.03518124826524033443366005160, 11.74844517042412344758384060474, 12.79805428786875630561963037970, 14.61299846699563525441950490354, 15.46514485473472644303531191982, 16.44607557155423495797566330432
