| L(s) = 1 | + (−1.72 + 0.677i)2-s + (1.05 − 0.975i)4-s + (−0.820 + 0.559i)5-s + (0.202 − 2.63i)7-s + (0.454 − 0.944i)8-s + (1.03 − 1.52i)10-s + (−0.669 + 4.44i)11-s + (0.491 + 0.391i)13-s + (1.43 + 4.68i)14-s + (−0.359 + 4.79i)16-s + (−7.67 + 2.36i)17-s + (0.358 + 0.206i)19-s + (−0.316 + 1.38i)20-s + (−1.85 − 8.11i)22-s + (2.22 − 7.21i)23-s + ⋯ |
| L(s) = 1 | + (−1.21 + 0.478i)2-s + (0.525 − 0.487i)4-s + (−0.366 + 0.250i)5-s + (0.0763 − 0.997i)7-s + (0.160 − 0.333i)8-s + (0.327 − 0.480i)10-s + (−0.201 + 1.33i)11-s + (0.136 + 0.108i)13-s + (0.384 + 1.25i)14-s + (−0.0898 + 1.19i)16-s + (−1.86 + 0.574i)17-s + (0.0821 + 0.0474i)19-s + (−0.0708 + 0.310i)20-s + (−0.394 − 1.73i)22-s + (0.463 − 1.50i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0121992 - 0.131653i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0121992 - 0.131653i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.202 + 2.63i)T \) |
| good | 2 | \( 1 + (1.72 - 0.677i)T + (1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (0.820 - 0.559i)T + (1.82 - 4.65i)T^{2} \) |
| 11 | \( 1 + (0.669 - 4.44i)T + (-10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (-0.491 - 0.391i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (7.67 - 2.36i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-0.358 - 0.206i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.22 + 7.21i)T + (-19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-0.523 - 0.119i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (7.47 - 4.31i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.17 + 4.79i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (4.20 + 2.02i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (6.50 - 3.13i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-2.32 - 5.92i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (1.40 + 1.51i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (10.2 + 6.98i)T + (21.5 + 54.9i)T^{2} \) |
| 61 | \( 1 + (-3.28 + 3.53i)T + (-4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-1.75 - 3.03i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.14 + 0.488i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-2.40 - 0.944i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (6.85 - 11.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.36 - 7.98i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (6.09 - 0.918i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 - 12.9iT - 97T^{2} \) |
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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
−10.99405039485617480284896759952, −10.67588882277450267344921875630, −9.680399345055092292302688468890, −8.837838725647246162982332602305, −7.959287419123358532259025372401, −6.97522822187680867943563350358, −6.74057849630825829711197263278, −4.75112077650554837986533349982, −3.83569147160427350005410994147, −1.81473220747313354341853401198,
0.12032281567365684721157881906, 1.92978071778703070801923235688, 3.20395348988360544351836548425, 4.89572349644597279093651270102, 5.90078982340978292657216410645, 7.27022371449380771755221618026, 8.419583489011863423002726273955, 8.769900716762302066950266547180, 9.541044247890682092412729061550, 10.70146025750958625549283692101
