| L(s) = 1 | − 0.384·2-s − 1.85·4-s + 3.33·5-s + 7-s + 1.48·8-s − 1.28·10-s + 0.515·11-s + 0.383·13-s − 0.384·14-s + 3.13·16-s + 5.06·17-s − 3.66·19-s − 6.17·20-s − 0.198·22-s − 5.75·23-s + 6.12·25-s − 0.147·26-s − 1.85·28-s − 5.53·29-s − 1.26·31-s − 4.16·32-s − 1.94·34-s + 3.33·35-s − 1.25·37-s + 1.40·38-s + 4.94·40-s − 3.87·41-s + ⋯ |
| L(s) = 1 | − 0.271·2-s − 0.926·4-s + 1.49·5-s + 0.377·7-s + 0.523·8-s − 0.405·10-s + 0.155·11-s + 0.106·13-s − 0.102·14-s + 0.783·16-s + 1.22·17-s − 0.840·19-s − 1.38·20-s − 0.0422·22-s − 1.20·23-s + 1.22·25-s − 0.0289·26-s − 0.350·28-s − 1.02·29-s − 0.226·31-s − 0.736·32-s − 0.334·34-s + 0.563·35-s − 0.206·37-s + 0.228·38-s + 0.781·40-s − 0.604·41-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 | \( 1 + 0.384T + 2T^{2} \) |
| 5 | \( 1 - 3.33T + 5T^{2} \) |
| 11 | \( 1 - 0.515T + 11T^{2} \) |
| 13 | \( 1 - 0.383T + 13T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 + 3.66T + 19T^{2} \) |
| 23 | \( 1 + 5.75T + 23T^{2} \) |
| 29 | \( 1 + 5.53T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 + 1.25T + 37T^{2} \) |
| 41 | \( 1 + 3.87T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 7.61T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 4.93T + 61T^{2} \) |
| 67 | \( 1 + 7.85T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 7.05T + 73T^{2} \) |
| 79 | \( 1 + 1.78T + 79T^{2} \) |
| 83 | \( 1 + 1.60T + 83T^{2} \) |
| 89 | \( 1 + 8.42T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.75185060080063118896894458834, −6.65780466454674244313835952648, −6.01506100940305911363804555045, −5.38652524051739079398191014587, −4.85582875054039541841765490201, −3.93065057084671343666466991401, −3.13075511951832829980291354711, −1.83021002051259873999196246404, −1.50071389221219043903915805729, 0,
1.50071389221219043903915805729, 1.83021002051259873999196246404, 3.13075511951832829980291354711, 3.93065057084671343666466991401, 4.85582875054039541841765490201, 5.38652524051739079398191014587, 6.01506100940305911363804555045, 6.65780466454674244313835952648, 7.75185060080063118896894458834
