| L(s) = 1 | − 0.775·2-s − 1.39·4-s − 0.833·5-s + 7-s + 2.63·8-s + 0.646·10-s − 0.821·11-s − 3.40·13-s − 0.775·14-s + 0.752·16-s + 2.50·17-s + 1.40·19-s + 1.16·20-s + 0.636·22-s − 0.947·23-s − 4.30·25-s + 2.64·26-s − 1.39·28-s − 1.78·29-s + 8.51·31-s − 5.85·32-s − 1.94·34-s − 0.833·35-s − 2.89·37-s − 1.08·38-s − 2.19·40-s − 2.30·41-s + ⋯ |
| L(s) = 1 | − 0.548·2-s − 0.699·4-s − 0.372·5-s + 0.377·7-s + 0.931·8-s + 0.204·10-s − 0.247·11-s − 0.945·13-s − 0.207·14-s + 0.188·16-s + 0.607·17-s + 0.321·19-s + 0.260·20-s + 0.135·22-s − 0.197·23-s − 0.861·25-s + 0.518·26-s − 0.264·28-s − 0.331·29-s + 1.53·31-s − 1.03·32-s − 0.332·34-s − 0.140·35-s − 0.475·37-s − 0.176·38-s − 0.347·40-s − 0.359·41-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 | \( 1 + 0.775T + 2T^{2} \) |
| 5 | \( 1 + 0.833T + 5T^{2} \) |
| 11 | \( 1 + 0.821T + 11T^{2} \) |
| 13 | \( 1 + 3.40T + 13T^{2} \) |
| 17 | \( 1 - 2.50T + 17T^{2} \) |
| 19 | \( 1 - 1.40T + 19T^{2} \) |
| 23 | \( 1 + 0.947T + 23T^{2} \) |
| 29 | \( 1 + 1.78T + 29T^{2} \) |
| 31 | \( 1 - 8.51T + 31T^{2} \) |
| 37 | \( 1 + 2.89T + 37T^{2} \) |
| 41 | \( 1 + 2.30T + 41T^{2} \) |
| 43 | \( 1 - 5.58T + 43T^{2} \) |
| 47 | \( 1 + 2.14T + 47T^{2} \) |
| 53 | \( 1 - 0.992T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 7.19T + 61T^{2} \) |
| 67 | \( 1 + 5.77T + 67T^{2} \) |
| 71 | \( 1 - 0.675T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 1.57T + 79T^{2} \) |
| 83 | \( 1 + 3.31T + 83T^{2} \) |
| 89 | \( 1 - 1.39T + 89T^{2} \) |
| 97 | \( 1 + 4.55T + 97T^{2} \) |
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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
−7.58651019510569743323467136871, −7.18010732324278823727386780392, −6.02675926150421205664299615333, −5.29209084670603204144157552963, −4.65816499992390512862396038310, −4.02756224829702310705355240356, −3.11892693909139720730687568259, −2.08626381099863407019707346032, −1.03979489818924682804878612492, 0,
1.03979489818924682804878612492, 2.08626381099863407019707346032, 3.11892693909139720730687568259, 4.02756224829702310705355240356, 4.65816499992390512862396038310, 5.29209084670603204144157552963, 6.02675926150421205664299615333, 7.18010732324278823727386780392, 7.58651019510569743323467136871
