| L(s) = 1 | + (1.36 − 4.20i)2-s + (−11.6 − 10.4i)3-s + (10.0 + 7.33i)4-s + (−98.1 + 31.8i)5-s + (−59.5 + 34.5i)6-s + (−5.71 + 7.86i)7-s + (158. − 115. i)8-s + (26.3 + 241. i)9-s + 455. i·10-s + (−294. + 273. i)11-s + (−40.7 − 190. i)12-s + (−752. − 244. i)13-s + (25.2 + 34.7i)14-s + (1.47e3 + 651. i)15-s + (−145. − 446. i)16-s + (−218. − 671. i)17-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.742i)2-s + (−0.744 − 0.667i)3-s + (0.315 + 0.229i)4-s + (−1.75 + 0.570i)5-s + (−0.675 + 0.391i)6-s + (−0.0440 + 0.0606i)7-s + (0.878 − 0.638i)8-s + (0.108 + 0.994i)9-s + 1.44i·10-s + (−0.732 + 0.680i)11-s + (−0.0817 − 0.381i)12-s + (−1.23 − 0.401i)13-s + (0.0344 + 0.0474i)14-s + (1.68 + 0.747i)15-s + (−0.141 − 0.435i)16-s + (−0.183 − 0.563i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.00523188 + 0.0102098i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00523188 + 0.0102098i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (11.6 + 10.4i)T \) |
| 11 | \( 1 + (294. - 273. i)T \) |
| good | 2 | \( 1 + (-1.36 + 4.20i)T + (-25.8 - 18.8i)T^{2} \) |
| 5 | \( 1 + (98.1 - 31.8i)T + (2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (5.71 - 7.86i)T + (-5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (752. + 244. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (218. + 671. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (1.02e3 + 1.40e3i)T + (-7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 1.91e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (4.40e3 + 3.19e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (805. - 2.47e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-533. - 387. i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (6.67e3 - 4.84e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.55e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-6.05e3 - 8.33e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.50e4 - 4.87e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (6.18e3 - 8.51e3i)T + (-2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.53e4 + 4.97e3i)T + (6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + 3.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (3.49e4 - 1.13e4i)T + (1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-1.75e4 + 2.41e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (6.52e4 + 2.12e4i)T + (2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (2.74e4 + 8.44e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + 9.08e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (4.77e4 - 1.46e5i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−15.13826356126068116098406096390, −13.08670203992304404679147570634, −12.15097894919520408330901833454, −11.45233510447210603071413830408, −10.48057199250643792296862920213, −7.63907159111764954925435506114, −7.16733493230173684501910958695, −4.57664169726833739298326656893, −2.67149038971736399018323137766, −0.00662474816679453884275208970,
4.13431795768529986998903513838, 5.37696580693881878959280049398, 7.09408529624803213844482194345, 8.390791728137614306661612180155, 10.48835315624731608496844081778, 11.49855812772073297328214571898, 12.56564071926179689543582279138, 14.76341336971477007370792457091, 15.37073988146802232220206215936, 16.53008465411110795098411166631
