| L(s) = 1 | − 2.82i·5-s − i·7-s − 6.29·11-s + 5.44·13-s − 0.317i·17-s + 4i·19-s − 0.317·23-s − 3.00·25-s + 5.19i·29-s + 8.79i·31-s − 2.82·35-s + 0.898·37-s + 3.46i·41-s − 5.44i·43-s − 12.5·47-s + ⋯ |
| L(s) = 1 | − 1.26i·5-s − 0.377i·7-s − 1.89·11-s + 1.51·13-s − 0.0770i·17-s + 0.917i·19-s − 0.0662·23-s − 0.600·25-s + 0.964i·29-s + 1.58i·31-s − 0.478·35-s + 0.147·37-s + 0.541i·41-s − 0.831i·43-s − 1.83·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.204166658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.204166658\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 11 | \( 1 + 6.29T + 11T^{2} \) |
| 13 | \( 1 - 5.44T + 13T^{2} \) |
| 17 | \( 1 + 0.317iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 0.317T + 23T^{2} \) |
| 29 | \( 1 - 5.19iT - 29T^{2} \) |
| 31 | \( 1 - 8.79iT - 31T^{2} \) |
| 37 | \( 1 - 0.898T + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 5.44iT - 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 - 8.02iT - 53T^{2} \) |
| 59 | \( 1 + 5.83T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 3.44iT - 67T^{2} \) |
| 71 | \( 1 - 2.51T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 - 10.7iT - 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.307668884408518059853978992817, −7.66863653321432416532871626260, −6.74956564538240923011850734199, −5.85295194255117244659352717593, −5.21327070216981639354709685472, −4.73376190314653348163919281211, −3.73550660506596879253320199686, −3.01734288593445113006318765298, −1.72929704757702201866203713460, −0.965905186833715964811239405347,
0.34702982273408692854889535084, 2.00881860080318880006025449059, 2.72797726114151154067711036227, 3.31076030984405050779621262057, 4.28471274146805054791621461475, 5.25749592089770479311734600102, 5.93815433090289711814827602119, 6.52054096196837691701134272595, 7.27335037640922203925380988946, 8.096356965501014358270991312905
