| L(s) = 1 | + (−1.37 − 0.327i)2-s + (−0.0831 − 1.73i)3-s + (1.78 + 0.902i)4-s + (−0.685 − 1.88i)5-s + (−0.453 + 2.40i)6-s + (−0.984 + 0.173i)7-s + (−2.15 − 1.82i)8-s + (−2.98 + 0.287i)9-s + (0.325 + 2.81i)10-s + (−1.63 − 0.596i)11-s + (1.41 − 3.16i)12-s + (2.68 + 2.25i)13-s + (1.41 + 0.0840i)14-s + (−3.20 + 1.34i)15-s + (2.37 + 3.22i)16-s + (−4.11 − 2.37i)17-s + ⋯ |
| L(s) = 1 | + (−0.972 − 0.231i)2-s + (−0.0479 − 0.998i)3-s + (0.892 + 0.451i)4-s + (−0.306 − 0.841i)5-s + (−0.184 + 0.982i)6-s + (−0.372 + 0.0656i)7-s + (−0.763 − 0.645i)8-s + (−0.995 + 0.0958i)9-s + (0.102 + 0.890i)10-s + (−0.493 − 0.179i)11-s + (0.407 − 0.913i)12-s + (0.745 + 0.625i)13-s + (0.377 + 0.0224i)14-s + (−0.826 + 0.346i)15-s + (0.592 + 0.805i)16-s + (−0.999 − 0.576i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0831 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0831 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000646195 + 0.000702333i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000646195 + 0.000702333i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 + 0.327i)T \) |
| 3 | \( 1 + (0.0831 + 1.73i)T \) |
| 7 | \( 1 + (0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (0.685 + 1.88i)T + (-3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (1.63 + 0.596i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.68 - 2.25i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (4.11 + 2.37i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.53 - 2.61i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.487 - 2.76i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.70 - 5.60i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (8.83 + 1.55i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.06 + 1.84i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.796 + 0.949i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.282 + 0.776i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.00999 - 0.0567i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 9.02iT - 53T^{2} \) |
| 59 | \( 1 + (6.96 - 2.53i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.697 - 3.95i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (3.21 - 3.83i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.83 - 3.18i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.20 - 12.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.42 - 7.66i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (7.59 - 6.37i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (6.38 - 3.68i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.64 - 0.599i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.75187296445821709020145746136, −9.433442838141870347525751153443, −8.671595540418620032776768762114, −8.279666847731450283476305010095, −7.17514133559179373676209737706, −6.52316412181096867057714761629, −5.48294756283122064118432412286, −3.91771069108575788776199159236, −2.54768167828424970655013597198, −1.41255009738623855753198408641,
0.00066037348851654385224626127, 2.42523324241622714704605795890, 3.38556141984634672273672132978, 4.64548041617712082499879647906, 6.00751999784538655467338888072, 6.55733916147073394081187335762, 7.70562700507343641929796335670, 8.570162158060362999589696781229, 9.234022785658917584768161909886, 10.27810687210720819765582339491
