| L(s) = 1 | + (−1.59 − 3.66i)2-s + (−10.9 + 11.6i)4-s + (14.6 − 14.6i)5-s − 24.0·7-s + (60.3 + 21.3i)8-s + (−76.8 − 30.2i)10-s + (−61.7 − 61.7i)11-s + (−37.5 − 37.5i)13-s + (38.2 + 88.1i)14-s + (−17.6 − 255. i)16-s − 96.8·17-s + (−156. + 156. i)19-s + (11.4 + 330. i)20-s + (−128. + 324. i)22-s − 959.·23-s + ⋯ |
| L(s) = 1 | + (−0.398 − 0.917i)2-s + (−0.682 + 0.731i)4-s + (0.584 − 0.584i)5-s − 0.490·7-s + (0.942 + 0.334i)8-s + (−0.768 − 0.302i)10-s + (−0.510 − 0.510i)11-s + (−0.222 − 0.222i)13-s + (0.195 + 0.449i)14-s + (−0.0689 − 0.997i)16-s − 0.335·17-s + (−0.434 + 0.434i)19-s + (0.0285 + 0.825i)20-s + (−0.264 + 0.671i)22-s − 1.81·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0516855 + 0.0834520i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0516855 + 0.0834520i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.59 + 3.66i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-14.6 + 14.6i)T - 625iT^{2} \) |
| 7 | \( 1 + 24.0T + 2.40e3T^{2} \) |
| 11 | \( 1 + (61.7 + 61.7i)T + 1.46e4iT^{2} \) |
| 13 | \( 1 + (37.5 + 37.5i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + 96.8T + 8.35e4T^{2} \) |
| 19 | \( 1 + (156. - 156. i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 + 959.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-350. - 350. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 - 237. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (560. - 560. i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 1.80e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-206. - 206. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 - 1.59e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (-2.23e3 + 2.23e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + (2.35e3 + 2.35e3i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (4.44e3 + 4.44e3i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + (3.99e3 - 3.99e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 + 4.92e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 2.65e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 8.79e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-228. + 228. i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 1.05e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.10e4T + 8.85e7T^{2} \) |
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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
−11.74788359170984458353865084997, −10.51887386451692101513566889827, −9.793602441771776064890298585520, −8.781572773278310870696147685108, −7.82312283912497528713365775316, −6.07496363793123329872357813182, −4.70761856189957202412182140224, −3.20536668140822338701977826809, −1.73432062376155469798870707688, −0.04258773369302950163090173301,
2.22409339070852393593260279311, 4.30680652366140566770832185534, 5.77772801421185832854202041610, 6.64425801700138507473139072715, 7.67034367189556987912040398793, 8.925581667097004057516077199388, 9.989242094920790157994469006173, 10.55290125120029277688636035940, 12.20908789755391901957667091083, 13.45407588856572436270636886985
