| L(s) = 1 | − i·5-s + 2.44i·7-s + 1.01·11-s − 2.24·13-s − 4.89i·19-s − 8.36·23-s − 25-s − 6i·29-s + 8.36i·31-s + 2.44·35-s − 6.24·37-s − 4.24i·41-s + 2.02i·43-s − 2.02·47-s + 1.00·49-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 0.925i·7-s + 0.305·11-s − 0.621·13-s − 1.12i·19-s − 1.74·23-s − 0.200·25-s − 1.11i·29-s + 1.50i·31-s + 0.414·35-s − 1.02·37-s − 0.662i·41-s + 0.309i·43-s − 0.295·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4874002616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4874002616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 11 | \( 1 - 1.01T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 8.36T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 8.36iT - 31T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 2.02iT - 43T^{2} \) |
| 47 | \( 1 + 2.02T + 47T^{2} \) |
| 53 | \( 1 + 8.48iT - 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 1.43iT - 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 16.2iT - 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.535808032090952880407281421936, −7.84841177465518556998775151130, −6.90213349003710129318314623291, −6.15310843997980585078875824479, −5.31812034001549240792779108864, −4.68356408746078087289598262140, −3.68522437288684132666782189977, −2.58419992277677218942028104397, −1.76149970523948463162571316459, −0.14822682806188445077048465039,
1.41651042501071630285084574842, 2.48649624927152110091861247242, 3.70388107184338122405855519743, 4.12360737979977434232859714462, 5.25788686727438812482485651186, 6.13380486649294854176352290668, 6.82039487871026122432010542311, 7.66928060849929382675790322851, 8.048811712640314072753764022588, 9.194882416298560044195147222022
