| L(s) = 1 | + (1.23 − 3.80i)2-s + 19.9·3-s + (−12.9 − 9.40i)4-s + (−63.9 − 46.4i)5-s + (24.6 − 75.9i)6-s + (−26.6 − 82.0i)7-s + (−51.7 + 37.6i)8-s + 155.·9-s + (−255. + 185. i)10-s + (−606. + 440. i)11-s + (−258. − 187. i)12-s + (108. − 332. i)13-s − 345.·14-s + (−1.27e3 − 927. i)15-s + (79.1 + 243. i)16-s + (954. − 693. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + 1.28·3-s + (−0.404 − 0.293i)4-s + (−1.14 − 0.830i)5-s + (0.279 − 0.861i)6-s + (−0.205 − 0.632i)7-s + (−0.286 + 0.207i)8-s + 0.640·9-s + (−0.808 + 0.587i)10-s + (−1.51 + 1.09i)11-s + (−0.518 − 0.376i)12-s + (0.177 − 0.546i)13-s − 0.470·14-s + (−1.46 − 1.06i)15-s + (0.0772 + 0.237i)16-s + (0.801 − 0.582i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0634i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0458987 - 1.44596i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0458987 - 1.44596i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 + 3.80i)T \) |
| 41 | \( 1 + (1.63e3 - 1.06e4i)T \) |
| good | 3 | \( 1 - 19.9T + 243T^{2} \) |
| 5 | \( 1 + (63.9 + 46.4i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (26.6 + 82.0i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 11 | \( 1 + (606. - 440. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-108. + 332. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-954. + 693. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (35.3 + 108. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-709. + 2.18e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (4.11e3 + 2.98e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-5.65e3 + 4.10e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (2.09e3 + 1.51e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 43 | \( 1 + (-1.43e3 + 4.40e3i)T + (-1.18e8 - 8.64e7i)T^{2} \) |
| 47 | \( 1 + (-1.27e3 + 3.93e3i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-2.16e4 - 1.57e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-9.53e3 + 2.93e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (5.60e3 + 1.72e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (5.50e4 + 4.00e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (1.86e4 - 1.35e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 - 6.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-2.82e4 - 8.68e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-1.01e4 - 7.40e3i)T + (2.65e9 + 8.16e9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.93040421934194565019412163708, −11.99026207135038790921730649268, −10.50940721590220584939897027734, −9.481628505055944037495247377687, −8.143370363251229370911901742483, −7.61896078768662617940687366007, −4.95401574109473872707506921732, −3.81162264731404943098036335377, −2.54394245018116008232431863515, −0.47242426055452827653356347346,
2.86739122122345580812433835121, 3.64428693355919533265399365654, 5.64445465201064560688017215658, 7.30914902889957261559041143317, 8.106359943373132189797669211157, 8.906301179787844474810217416197, 10.53800668073147952548545208781, 11.83610341946828135140140156946, 13.19974208098773726575922796756, 14.07679720429733601458062208924
