L(s) = 1 | + (5.64 + 0.294i)2-s + (−20.2 − 20.2i)3-s + (31.8 + 3.32i)4-s + (16.5 − 53.4i)5-s + (−108. − 120. i)6-s + (9.02 − 9.02i)7-s + (178. + 28.1i)8-s + 574. i·9-s + (109. − 296. i)10-s + 200. i·11-s + (−576. − 710. i)12-s + (362. − 362. i)13-s + (53.6 − 48.3i)14-s + (−1.41e3 + 745. i)15-s + (1.00e3 + 211. i)16-s + (465. + 465. i)17-s + ⋯ |
L(s) = 1 | + (0.998 + 0.0520i)2-s + (−1.29 − 1.29i)3-s + (0.994 + 0.104i)4-s + (0.295 − 0.955i)5-s + (−1.22 − 1.36i)6-s + (0.0695 − 0.0695i)7-s + (0.987 + 0.155i)8-s + 2.36i·9-s + (0.345 − 0.938i)10-s + 0.500i·11-s + (−1.15 − 1.42i)12-s + (0.595 − 0.595i)13-s + (0.0731 − 0.0658i)14-s + (−1.62 + 0.855i)15-s + (0.978 + 0.206i)16-s + (0.391 + 0.391i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.35184 - 1.16395i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35184 - 1.16395i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-5.64 - 0.294i)T \) |
| 5 | \( 1 + (-16.5 + 53.4i)T \) |
good | 3 | \( 1 + (20.2 + 20.2i)T + 243iT^{2} \) |
| 7 | \( 1 + (-9.02 + 9.02i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 - 200. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-362. + 362. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-465. - 465. i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 - 582.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (285. + 285. i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 - 1.27e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.26e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (6.27e3 + 6.27e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 + 1.16e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.19e4 - 1.19e4i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (4.13e3 - 4.13e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.37e4 - 1.37e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + 3.81e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.00e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (3.07e3 - 3.07e3i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 + 4.94e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.54e4 + 2.54e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 9.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (3.98e4 + 3.98e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 6.51e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (4.46e4 + 4.46e4i)T + 8.58e9iT^{2} \) |
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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
−16.99534606732205062320550190970, −15.93899668108470059973037613185, −13.85290379660974737897865731306, −12.75850998728749872964207602160, −12.18599576570234483797703823406, −10.77106289900906988519373566253, −7.75106745869622577388887822310, −6.20649209077934850833552959765, −5.07158269768732575135225895777, −1.41544671053104563461674453739,
3.66829310731881366334750895765, 5.39460481386876125967999445822, 6.55979766933551388366057825302, 9.943528708825998133773606446931, 11.05307384530100316877085757151, 11.81224617897902553294314351944, 13.86160938705228380713483928677, 15.14596668108327034953357550140, 16.07087669544534784121465565853, 17.14566458657299701233652585242