| L(s) = 1 | − 0.554·2-s − 3-s − 1.69·4-s − 2.60·5-s + 0.554·6-s + 2.71·7-s + 2.04·8-s + 9-s + 1.44·10-s + 3.78·11-s + 1.69·12-s − 2.37·13-s − 1.50·14-s + 2.60·15-s + 2.24·16-s + 17-s − 0.554·18-s + 5.06·19-s + 4.40·20-s − 2.71·21-s − 2.09·22-s − 4.47·23-s − 2.04·24-s + 1.76·25-s + 1.31·26-s − 27-s − 4.59·28-s + ⋯ |
| L(s) = 1 | − 0.392·2-s − 0.577·3-s − 0.846·4-s − 1.16·5-s + 0.226·6-s + 1.02·7-s + 0.723·8-s + 0.333·9-s + 0.456·10-s + 1.14·11-s + 0.488·12-s − 0.658·13-s − 0.402·14-s + 0.671·15-s + 0.562·16-s + 0.242·17-s − 0.130·18-s + 1.16·19-s + 0.984·20-s − 0.592·21-s − 0.447·22-s − 0.933·23-s − 0.417·24-s + 0.352·25-s + 0.258·26-s − 0.192·27-s − 0.868·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 + 0.554T + 2T^{2} \) |
| 5 | \( 1 + 2.60T + 5T^{2} \) |
| 7 | \( 1 - 2.71T + 7T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 13 | \( 1 + 2.37T + 13T^{2} \) |
| 19 | \( 1 - 5.06T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 7.08T + 29T^{2} \) |
| 31 | \( 1 + 6.71T + 31T^{2} \) |
| 37 | \( 1 - 8.80T + 37T^{2} \) |
| 41 | \( 1 - 4.09T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 2.56T + 53T^{2} \) |
| 59 | \( 1 - 1.26T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 1.50T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 0.155T + 73T^{2} \) |
| 79 | \( 1 + 9.11T + 79T^{2} \) |
| 83 | \( 1 + 0.232T + 83T^{2} \) |
| 89 | \( 1 + 9.44T + 89T^{2} \) |
| 97 | \( 1 - 1.00T + 97T^{2} \) |
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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
−7.61301545733513499423710741243, −7.15989150052017138548019655520, −5.96009448870050880224932857888, −5.30340501004698557177051717278, −4.57536633218455525377534258272, −4.04875390461812423342309461303, −3.45531297837143029375219704566, −1.86200939412863019133405121434, −1.01638685346333011730286127119, 0,
1.01638685346333011730286127119, 1.86200939412863019133405121434, 3.45531297837143029375219704566, 4.04875390461812423342309461303, 4.57536633218455525377534258272, 5.30340501004698557177051717278, 5.96009448870050880224932857888, 7.15989150052017138548019655520, 7.61301545733513499423710741243
