| L(s) = 1 | + 1.73·5-s + (2.5 + 0.866i)7-s + 3i·11-s − 6.92i·13-s + 6.92·17-s + 3.46i·19-s − 6i·23-s − 2.00·25-s − 6i·29-s − 5.19i·31-s + (4.33 + 1.49i)35-s − 2·37-s + 3.46·41-s + 2·43-s + 3.46·47-s + ⋯ |
| L(s) = 1 | + 0.774·5-s + (0.944 + 0.327i)7-s + 0.904i·11-s − 1.92i·13-s + 1.68·17-s + 0.794i·19-s − 1.25i·23-s − 0.400·25-s − 1.11i·29-s − 0.933i·31-s + (0.731 + 0.253i)35-s − 0.328·37-s + 0.541·41-s + 0.304·43-s + 0.505·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.573199985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.573199985\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
| good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 13 | \( 1 + 6.92iT - 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 5.19iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 - 12.1iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 1.73T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 12.1iT - 97T^{2} \) |
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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
−8.565493392533570112972644500546, −7.73766266852820200923132735705, −7.60751983687230112749821530544, −6.03214471779589205005105606008, −5.74689395077437169558879317679, −4.96970325756372227392625289919, −4.01934717521765141056227569094, −2.83753524379660739225850369003, −2.04604104851733677871918777163, −0.931553221629601885976245899392,
1.22565207443638167004964372409, 1.87890999684108893766373150336, 3.16859811553518843344275301406, 4.04490367473859350494817356454, 5.06523659664105468867235254468, 5.58537974474968427434630323766, 6.51441214592387678420953929037, 7.27580612410029086330069862267, 7.972988418957966234815220365323, 8.973918517254923365092155298729
