| L(s) = 1 | + 2.43·2-s − 3-s + 3.92·4-s + 1.95·5-s − 2.43·6-s − 3.76·7-s + 4.68·8-s + 9-s + 4.76·10-s − 2.73·11-s − 3.92·12-s − 13-s − 9.17·14-s − 1.95·15-s + 3.56·16-s − 0.458·17-s + 2.43·18-s − 7.22·19-s + 7.68·20-s + 3.76·21-s − 6.66·22-s − 1.13·23-s − 4.68·24-s − 1.16·25-s − 2.43·26-s − 27-s − 14.7·28-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 0.577·3-s + 1.96·4-s + 0.875·5-s − 0.993·6-s − 1.42·7-s + 1.65·8-s + 0.333·9-s + 1.50·10-s − 0.825·11-s − 1.13·12-s − 0.277·13-s − 2.45·14-s − 0.505·15-s + 0.890·16-s − 0.111·17-s + 0.573·18-s − 1.65·19-s + 1.71·20-s + 0.822·21-s − 1.42·22-s − 0.236·23-s − 0.957·24-s − 0.233·25-s − 0.477·26-s − 0.192·27-s − 2.79·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 5 | \( 1 - 1.95T + 5T^{2} \) |
| 7 | \( 1 + 3.76T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 17 | \( 1 + 0.458T + 17T^{2} \) |
| 19 | \( 1 + 7.22T + 19T^{2} \) |
| 23 | \( 1 + 1.13T + 23T^{2} \) |
| 29 | \( 1 + 0.00282T + 29T^{2} \) |
| 31 | \( 1 + 3.50T + 31T^{2} \) |
| 37 | \( 1 + 3.90T + 37T^{2} \) |
| 41 | \( 1 - 5.89T + 41T^{2} \) |
| 43 | \( 1 + 2.09T + 43T^{2} \) |
| 47 | \( 1 - 7.67T + 47T^{2} \) |
| 53 | \( 1 + 1.62T + 53T^{2} \) |
| 59 | \( 1 + 5.92T + 59T^{2} \) |
| 61 | \( 1 + 1.77T + 61T^{2} \) |
| 67 | \( 1 + 7.70T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 9.18T + 73T^{2} \) |
| 79 | \( 1 - 8.08T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 7.62T + 97T^{2} \) |
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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
−7.68011904186293875401630069327, −6.83256526909172031966643789076, −6.26793855383399372016820642426, −5.85961458972757946674842262667, −5.17774379620201160515720730261, −4.33875699448533033090086901428, −3.57045876556600687345970237902, −2.63043332726232031348877459270, −2.00391580931453212112892096295, 0,
2.00391580931453212112892096295, 2.63043332726232031348877459270, 3.57045876556600687345970237902, 4.33875699448533033090086901428, 5.17774379620201160515720730261, 5.85961458972757946674842262667, 6.26793855383399372016820642426, 6.83256526909172031966643789076, 7.68011904186293875401630069327
