| L(s) = 1 | − 1.57·2-s + 0.465·4-s + 2.53·5-s − 7-s + 2.40·8-s − 3.98·10-s − 4.51·11-s + 1.53·13-s + 1.57·14-s − 4.71·16-s + 5.77·17-s + 2.92·19-s + 1.18·20-s + 7.08·22-s + 0.932·23-s + 1.45·25-s − 2.41·26-s − 0.465·28-s + 4.25·29-s − 7.25·31-s + 2.58·32-s − 9.06·34-s − 2.53·35-s − 11.7·37-s − 4.59·38-s + 6.12·40-s − 3.89·41-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 0.232·4-s + 1.13·5-s − 0.377·7-s + 0.852·8-s − 1.26·10-s − 1.36·11-s + 0.425·13-s + 0.419·14-s − 1.17·16-s + 1.40·17-s + 0.670·19-s + 0.264·20-s + 1.51·22-s + 0.194·23-s + 0.290·25-s − 0.472·26-s − 0.0878·28-s + 0.790·29-s − 1.30·31-s + 0.456·32-s − 1.55·34-s − 0.429·35-s − 1.92·37-s − 0.744·38-s + 0.967·40-s − 0.609·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 1.57T + 2T^{2} \) |
| 5 | \( 1 - 2.53T + 5T^{2} \) |
| 11 | \( 1 + 4.51T + 11T^{2} \) |
| 13 | \( 1 - 1.53T + 13T^{2} \) |
| 17 | \( 1 - 5.77T + 17T^{2} \) |
| 19 | \( 1 - 2.92T + 19T^{2} \) |
| 23 | \( 1 - 0.932T + 23T^{2} \) |
| 29 | \( 1 - 4.25T + 29T^{2} \) |
| 31 | \( 1 + 7.25T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 3.89T + 41T^{2} \) |
| 43 | \( 1 + 6.88T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 7.51T + 59T^{2} \) |
| 61 | \( 1 - 1.82T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 4.84T + 71T^{2} \) |
| 73 | \( 1 - 5.80T + 73T^{2} \) |
| 79 | \( 1 + 5.73T + 79T^{2} \) |
| 83 | \( 1 - 3.32T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 1.34T + 97T^{2} \) |
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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
−7.50845171910019502974329924662, −7.11055510008380623547868365382, −6.10835104334967734459259764459, −5.34412718091597832009757285090, −5.06507248273428243200410733384, −3.70189831070230714680002648889, −2.91890533198574643036854489134, −1.93631843970686485926904219852, −1.20534311465344561608623435744, 0,
1.20534311465344561608623435744, 1.93631843970686485926904219852, 2.91890533198574643036854489134, 3.70189831070230714680002648889, 5.06507248273428243200410733384, 5.34412718091597832009757285090, 6.10835104334967734459259764459, 7.11055510008380623547868365382, 7.50845171910019502974329924662
