| L(s) = 1 | + 1.29·2-s + 1.54·3-s − 0.316·4-s − 5-s + 1.99·6-s − 3.00·8-s − 0.627·9-s − 1.29·10-s − 11-s − 0.487·12-s + 6.24·13-s − 1.54·15-s − 3.26·16-s − 5.91·17-s − 0.814·18-s − 1.35·19-s + 0.316·20-s − 1.29·22-s − 2.02·23-s − 4.62·24-s + 25-s + 8.10·26-s − 5.58·27-s − 2.96·29-s − 1.99·30-s − 4.99·31-s + 1.77·32-s + ⋯ |
| L(s) = 1 | + 0.917·2-s + 0.889·3-s − 0.158·4-s − 0.447·5-s + 0.815·6-s − 1.06·8-s − 0.209·9-s − 0.410·10-s − 0.301·11-s − 0.140·12-s + 1.73·13-s − 0.397·15-s − 0.816·16-s − 1.43·17-s − 0.191·18-s − 0.310·19-s + 0.0707·20-s − 0.276·22-s − 0.422·23-s − 0.945·24-s + 0.200·25-s + 1.58·26-s − 1.07·27-s − 0.550·29-s − 0.364·30-s − 0.897·31-s + 0.313·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.29T + 2T^{2} \) |
| 3 | \( 1 - 1.54T + 3T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 19 | \( 1 + 1.35T + 19T^{2} \) |
| 23 | \( 1 + 2.02T + 23T^{2} \) |
| 29 | \( 1 + 2.96T + 29T^{2} \) |
| 31 | \( 1 + 4.99T + 31T^{2} \) |
| 37 | \( 1 + 1.31T + 37T^{2} \) |
| 41 | \( 1 + 8.18T + 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 + 9.70T + 47T^{2} \) |
| 53 | \( 1 + 2.59T + 53T^{2} \) |
| 59 | \( 1 - 0.0661T + 59T^{2} \) |
| 61 | \( 1 - 7.57T + 61T^{2} \) |
| 67 | \( 1 - 1.70T + 67T^{2} \) |
| 71 | \( 1 + 3.74T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 0.240T + 79T^{2} \) |
| 83 | \( 1 - 9.08T + 83T^{2} \) |
| 89 | \( 1 - 3.34T + 89T^{2} \) |
| 97 | \( 1 - 1.30T + 97T^{2} \) |
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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
−8.653886520037807192940677860362, −7.908141570674229511682847724142, −6.77128251417607461543023012707, −6.05438566405230997571196957250, −5.25183982328858333212355365429, −4.22045877313898228651939988920, −3.70050832443028829569983403620, −2.99603826907860840292470022476, −1.90407072163281678524231327701, 0,
1.90407072163281678524231327701, 2.99603826907860840292470022476, 3.70050832443028829569983403620, 4.22045877313898228651939988920, 5.25183982328858333212355365429, 6.05438566405230997571196957250, 6.77128251417607461543023012707, 7.908141570674229511682847724142, 8.653886520037807192940677860362
