| L(s) = 1 | + (3.80 + 1.23i)2-s + (12.0 − 16.5i)3-s + (12.9 + 9.40i)4-s + (−33.0 − 45.0i)5-s + (66.1 − 48.0i)6-s − 103. i·7-s + (37.6 + 51.7i)8-s + (−54.1 − 166. i)9-s + (−70.2 − 212. i)10-s + (−4.59 + 14.1i)11-s + (311. − 101. i)12-s + (−100. + 32.6i)13-s + (128. − 395. i)14-s + (−1.14e3 + 6.01i)15-s + (79.1 + 243. i)16-s + (343. + 472. i)17-s + ⋯ |
| L(s) = 1 | + (0.672 + 0.218i)2-s + (0.771 − 1.06i)3-s + (0.404 + 0.293i)4-s + (−0.592 − 0.805i)5-s + (0.750 − 0.545i)6-s − 0.801i·7-s + (0.207 + 0.286i)8-s + (−0.222 − 0.686i)9-s + (−0.222 − 0.671i)10-s + (−0.0114 + 0.0352i)11-s + (0.623 − 0.202i)12-s + (−0.165 + 0.0536i)13-s + (0.175 − 0.538i)14-s + (−1.31 + 0.00690i)15-s + (0.0772 + 0.237i)16-s + (0.288 + 0.396i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.20739 - 1.69370i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.20739 - 1.69370i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.80 - 1.23i)T \) |
| 5 | \( 1 + (33.0 + 45.0i)T \) |
| good | 3 | \( 1 + (-12.0 + 16.5i)T + (-75.0 - 231. i)T^{2} \) |
| 7 | \( 1 + 103. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (4.59 - 14.1i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (100. - 32.6i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-343. - 472. i)T + (-4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.97e3 + 1.43e3i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-4.00e3 - 1.30e3i)T + (5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (1.83e3 + 1.33e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (6.28e3 - 4.56e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (5.28e3 - 1.71e3i)T + (5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-328. - 1.01e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.95e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-8.30e3 + 1.14e4i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.55e4 - 2.13e4i)T + (-1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.21e3 + 3.73e3i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.19e4 + 3.68e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (3.31e4 + 4.56e4i)T + (-4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-2.81e4 - 2.04e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.31e4 - 4.26e3i)T + (1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (2.38e4 + 1.73e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (3.39e4 + 4.66e4i)T + (-1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (2.40e3 - 7.40e3i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (7.17e4 - 9.87e4i)T + (-2.65e9 - 8.16e9i)T^{2} \) |
| 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
−14.01803757382639474644834164489, −13.22294179601076101857331446624, −12.46744447256955336838263243448, −11.16666336960070061840442419969, −9.090149745299949521523116994373, −7.74101481842885917360196442006, −7.07780635844514933485561891359, −5.00065457061097143830042312562, −3.31756600777403225941732719910, −1.23266732511421789400539042421,
2.78852337767146832688698935869, 3.78622594446057185294300576135, 5.40172191407016712751506426091, 7.30514438134050311765834832740, 8.929048553669404271288820344714, 10.11534739681798708309137096187, 11.27684864847824722902476406674, 12.40896855528353782683104851027, 14.06234891762890786238006474096, 14.83514584522423395625382649781
