| L(s) = 1 | − 2.11·2-s + 2.49·4-s − 3.20·5-s + 7-s − 1.04·8-s + 6.79·10-s − 4.37·11-s + 3.90·13-s − 2.11·14-s − 2.76·16-s + 3.95·17-s − 4.13·19-s − 7.99·20-s + 9.28·22-s − 5.86·23-s + 5.28·25-s − 8.27·26-s + 2.49·28-s − 1.39·29-s − 0.976·31-s + 7.96·32-s − 8.39·34-s − 3.20·35-s + 2.06·37-s + 8.77·38-s + 3.35·40-s + 7.92·41-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.24·4-s − 1.43·5-s + 0.377·7-s − 0.369·8-s + 2.14·10-s − 1.32·11-s + 1.08·13-s − 0.566·14-s − 0.692·16-s + 0.960·17-s − 0.949·19-s − 1.78·20-s + 1.97·22-s − 1.22·23-s + 1.05·25-s − 1.62·26-s + 0.471·28-s − 0.258·29-s − 0.175·31-s + 1.40·32-s − 1.43·34-s − 0.542·35-s + 0.338·37-s + 1.42·38-s + 0.530·40-s + 1.23·41-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 | \( 1 + 2.11T + 2T^{2} \) |
| 5 | \( 1 + 3.20T + 5T^{2} \) |
| 11 | \( 1 + 4.37T + 11T^{2} \) |
| 13 | \( 1 - 3.90T + 13T^{2} \) |
| 17 | \( 1 - 3.95T + 17T^{2} \) |
| 19 | \( 1 + 4.13T + 19T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 + 0.976T + 31T^{2} \) |
| 37 | \( 1 - 2.06T + 37T^{2} \) |
| 41 | \( 1 - 7.92T + 41T^{2} \) |
| 43 | \( 1 - 7.54T + 43T^{2} \) |
| 47 | \( 1 + 0.247T + 47T^{2} \) |
| 53 | \( 1 + 4.76T + 53T^{2} \) |
| 59 | \( 1 - 7.06T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 + 3.83T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 - 8.46T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 0.301T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−7.65971384559870757479121601287, −7.38722653699170111268365932137, −6.29176409648895069511069875327, −5.56323600932049352029375596007, −4.45042632758171157114449501963, −3.94517266566929516880872352823, −2.91931513459708662342804501992, −1.96674659692090982513028736781, −0.880094081354726456716004195305, 0,
0.880094081354726456716004195305, 1.96674659692090982513028736781, 2.91931513459708662342804501992, 3.94517266566929516880872352823, 4.45042632758171157114449501963, 5.56323600932049352029375596007, 6.29176409648895069511069875327, 7.38722653699170111268365932137, 7.65971384559870757479121601287
