| L(s) = 1 | + 0.456·2-s − 1.79·4-s + 4.37·5-s − 7-s − 1.73·8-s + 1.99·10-s + 2.64·11-s + 4·13-s − 0.456·14-s + 2.79·16-s − 3.46·17-s − 3.58·19-s − 7.84·20-s + 1.20·22-s + 3.46·23-s + 14.1·25-s + 1.82·26-s + 1.79·28-s − 1.82·29-s + 9.16·31-s + 4.73·32-s − 1.58·34-s − 4.37·35-s + 3·37-s − 1.63·38-s − 7.58·40-s − 4.37·41-s + ⋯ |
| L(s) = 1 | + 0.323·2-s − 0.895·4-s + 1.95·5-s − 0.377·7-s − 0.612·8-s + 0.632·10-s + 0.797·11-s + 1.10·13-s − 0.122·14-s + 0.697·16-s − 0.840·17-s − 0.821·19-s − 1.75·20-s + 0.257·22-s + 0.722·23-s + 2.83·25-s + 0.358·26-s + 0.338·28-s − 0.339·29-s + 1.64·31-s + 0.837·32-s − 0.271·34-s − 0.739·35-s + 0.493·37-s − 0.265·38-s − 1.19·40-s − 0.683·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.934762486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.934762486\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - 0.456T + 2T^{2} \) |
| 5 | \( 1 - 4.37T + 5T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 1.82T + 29T^{2} \) |
| 31 | \( 1 - 9.16T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 4.37T + 41T^{2} \) |
| 43 | \( 1 + 8.58T + 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 2.41T + 61T^{2} \) |
| 67 | \( 1 + 0.582T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 3.16T + 73T^{2} \) |
| 79 | \( 1 + 8.58T + 79T^{2} \) |
| 83 | \( 1 + 6.20T + 83T^{2} \) |
| 89 | \( 1 + 8.75T + 89T^{2} \) |
| 97 | \( 1 - 7.58T + 97T^{2} \) |
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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
−10.44929116779862199668722574243, −9.851566738080430906038914415557, −8.936373241151046766073544684886, −8.632353312900625844915754984333, −6.59538379825885276397421169848, −6.22056816477371260095811627135, −5.23905951522094044400827279745, −4.20329461510527309036142478029, −2.85869553973039081867016434272, −1.39233024194959116741180448877,
1.39233024194959116741180448877, 2.85869553973039081867016434272, 4.20329461510527309036142478029, 5.23905951522094044400827279745, 6.22056816477371260095811627135, 6.59538379825885276397421169848, 8.632353312900625844915754984333, 8.936373241151046766073544684886, 9.851566738080430906038914415557, 10.44929116779862199668722574243
