| L(s) = 1 | + (−0.840 − 0.330i)2-s + (−1.12 + 5.07i)3-s + (−5.26 − 4.88i)4-s + (−8.71 − 5.94i)5-s + (2.62 − 3.89i)6-s + (17.4 − 6.25i)7-s + (5.95 + 12.3i)8-s + (−24.4 − 11.4i)9-s + (5.36 + 7.87i)10-s + (5.33 + 35.4i)11-s + (30.7 − 21.1i)12-s + (28.1 − 22.4i)13-s + (−16.7 − 0.492i)14-s + (39.9 − 37.5i)15-s + (3.36 + 44.9i)16-s + (100. + 31.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.297 − 0.116i)2-s + (−0.217 + 0.976i)3-s + (−0.658 − 0.610i)4-s + (−0.779 − 0.531i)5-s + (0.178 − 0.264i)6-s + (0.941 − 0.337i)7-s + (0.263 + 0.546i)8-s + (−0.905 − 0.424i)9-s + (0.169 + 0.249i)10-s + (0.146 + 0.970i)11-s + (0.739 − 0.509i)12-s + (0.601 − 0.479i)13-s + (−0.319 − 0.00940i)14-s + (0.688 − 0.645i)15-s + (0.0526 + 0.702i)16-s + (1.43 + 0.443i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 - 0.680i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.733 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.929689 + 0.364778i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.929689 + 0.364778i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.12 - 5.07i)T \) |
| 7 | \( 1 + (-17.4 + 6.25i)T \) |
| good | 2 | \( 1 + (0.840 + 0.330i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (8.71 + 5.94i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (-5.33 - 35.4i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (-28.1 + 22.4i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-100. - 31.0i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (83.8 - 48.4i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-49.4 - 160. i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-7.72 + 1.76i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-259. - 149. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-230. + 213. i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (250. - 120. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-338. - 163. i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-5.92 + 15.0i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-445. + 480. i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (495. - 337. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (329. + 355. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (438. - 760. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-23.0 - 5.25i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-293. + 115. i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-265. - 459. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (603. - 756. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (507. + 76.4i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 - 648. iT - 9.12e5T^{2} \) |
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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
−12.50061011503379533061863285299, −11.48855301909498798673148052316, −10.50727442558075134379848953691, −9.801917165496663714246007645934, −8.577227168376752738300131851263, −7.88340260347387296668967667104, −5.74823353377788363796137503058, −4.71952267415221016406338691250, −3.91710207677702158541115744691, −1.12887053466850924911026499431,
0.74545028843507656107831096369, 2.94733832959174007975009000150, 4.51317687114711421081680140335, 6.13519273673680263012913782404, 7.40707564736965370031022045504, 8.198936743293895161680904205031, 8.844978099196597114270525995637, 10.73305973972009843561336665997, 11.63920560657864466360390942943, 12.28289439088686927256155188860
