| L(s) = 1 | + 1.61·2-s + 0.618·4-s − 0.618·5-s + 7-s − 2.23·8-s − 1.00·10-s − 2.23·11-s − 3·13-s + 1.61·14-s − 4.85·16-s + 7.47·17-s + 1.85·19-s − 0.381·20-s − 3.61·22-s − 2.61·23-s − 4.61·25-s − 4.85·26-s + 0.618·28-s − 1.14·29-s − 7.38·31-s − 3.38·32-s + 12.0·34-s − 0.618·35-s − 5.70·37-s + 3·38-s + 1.38·40-s − 41-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.309·4-s − 0.276·5-s + 0.377·7-s − 0.790·8-s − 0.316·10-s − 0.674·11-s − 0.832·13-s + 0.432·14-s − 1.21·16-s + 1.81·17-s + 0.425·19-s − 0.0854·20-s − 0.771·22-s − 0.545·23-s − 0.923·25-s − 0.951·26-s + 0.116·28-s − 0.212·29-s − 1.32·31-s − 0.597·32-s + 2.07·34-s − 0.104·35-s − 0.938·37-s + 0.486·38-s + 0.218·40-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 - 7.47T + 17T^{2} \) |
| 19 | \( 1 - 1.85T + 19T^{2} \) |
| 23 | \( 1 + 2.61T + 23T^{2} \) |
| 29 | \( 1 + 1.14T + 29T^{2} \) |
| 31 | \( 1 + 7.38T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 43 | \( 1 + 9.94T + 43T^{2} \) |
| 47 | \( 1 + 1.23T + 47T^{2} \) |
| 53 | \( 1 - 2.90T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 + 11T + 61T^{2} \) |
| 67 | \( 1 - 6.61T + 67T^{2} \) |
| 71 | \( 1 - 4.23T + 71T^{2} \) |
| 73 | \( 1 + 15T + 73T^{2} \) |
| 79 | \( 1 - 5.70T + 79T^{2} \) |
| 83 | \( 1 - 5.94T + 83T^{2} \) |
| 89 | \( 1 + 5.56T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−8.280982006723427800265202144313, −7.69905007562915041416456307773, −6.94623546880007142732239382620, −5.69494429667699046616300412118, −5.41277829155777960192016593523, −4.60501961806929731741363899904, −3.64613652986103390694450751576, −3.04311338295008725647001955626, −1.82118342276237688063006704684, 0,
1.82118342276237688063006704684, 3.04311338295008725647001955626, 3.64613652986103390694450751576, 4.60501961806929731741363899904, 5.41277829155777960192016593523, 5.69494429667699046616300412118, 6.94623546880007142732239382620, 7.69905007562915041416456307773, 8.280982006723427800265202144313
