| L(s) = 1 | + (−0.784 − 1.17i)2-s + (−1.61 − 0.634i)3-s + (−0.767 + 1.84i)4-s + (0.557 − 1.53i)5-s + (0.518 + 2.39i)6-s + (0.984 + 0.173i)7-s + (2.77 − 0.546i)8-s + (2.19 + 2.04i)9-s + (−2.24 + 0.546i)10-s + (−3.97 + 1.44i)11-s + (2.40 − 2.48i)12-s + (1.96 − 1.65i)13-s + (−0.568 − 1.29i)14-s + (−1.87 + 2.11i)15-s + (−2.82 − 2.83i)16-s + (−6.64 + 3.83i)17-s + ⋯ |
| L(s) = 1 | + (−0.555 − 0.831i)2-s + (−0.930 − 0.366i)3-s + (−0.383 + 0.923i)4-s + (0.249 − 0.685i)5-s + (0.211 + 0.977i)6-s + (0.372 + 0.0656i)7-s + (0.981 − 0.193i)8-s + (0.731 + 0.681i)9-s + (−0.708 + 0.172i)10-s + (−1.19 + 0.435i)11-s + (0.695 − 0.718i)12-s + (0.545 − 0.457i)13-s + (−0.152 − 0.346i)14-s + (−0.483 + 0.546i)15-s + (−0.705 − 0.708i)16-s + (−1.61 + 0.930i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.732614 - 0.131833i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.732614 - 0.131833i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.784 + 1.17i)T \) |
| 3 | \( 1 + (1.61 + 0.634i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (-0.557 + 1.53i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (3.97 - 1.44i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.96 + 1.65i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (6.64 - 3.83i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-7.29 - 4.21i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.09 - 6.22i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.767 - 0.915i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.483 + 0.0852i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.215 + 0.373i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.21 - 6.21i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.678 + 1.86i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.12 + 12.0i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 9.24iT - 53T^{2} \) |
| 59 | \( 1 + (-11.9 - 4.33i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.305 + 1.73i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.72 - 5.63i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.796 - 1.38i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.937 + 1.62i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.09 - 8.45i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.48 - 7.12i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-5.64 - 3.26i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.36 + 0.859i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.36603687376875795783692070681, −9.697276138211770794934866897865, −8.582337715106220567307063202621, −7.86802200651627305170604918325, −7.03040622151333445602395020300, −5.54853137541115270688660612360, −5.01535299617058252765362753659, −3.76411887494700995491866694679, −2.13381616767816533562586874847, −1.11377566685632553758156766488,
0.63223049384401363882108232422, 2.61365259651527246723077380469, 4.48027302637640060163924908376, 5.13078146103586322587324351437, 6.12091890034629647185521089558, 6.85715212284744688629900681860, 7.55751619802304474875197445714, 8.816118410598663479364574793798, 9.461831110825546224760002009478, 10.55829342308193508314132095898
