| L(s) = 1 | − 1.94·2-s + 1.78·4-s − 1.34·5-s + 7-s + 0.426·8-s + 2.61·10-s + 0.671·11-s − 2.12·13-s − 1.94·14-s − 4.39·16-s − 1.91·17-s − 4.48·19-s − 2.39·20-s − 1.30·22-s + 4.40·23-s − 3.19·25-s + 4.12·26-s + 1.78·28-s + 2.36·29-s + 1.43·31-s + 7.68·32-s + 3.71·34-s − 1.34·35-s + 5.60·37-s + 8.71·38-s − 0.572·40-s + 4.45·41-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 0.890·4-s − 0.600·5-s + 0.377·7-s + 0.150·8-s + 0.825·10-s + 0.202·11-s − 0.589·13-s − 0.519·14-s − 1.09·16-s − 0.463·17-s − 1.02·19-s − 0.534·20-s − 0.278·22-s + 0.918·23-s − 0.639·25-s + 0.809·26-s + 0.336·28-s + 0.438·29-s + 0.258·31-s + 1.35·32-s + 0.637·34-s − 0.226·35-s + 0.921·37-s + 1.41·38-s − 0.0905·40-s + 0.696·41-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 | \( 1 + 1.94T + 2T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 11 | \( 1 - 0.671T + 11T^{2} \) |
| 13 | \( 1 + 2.12T + 13T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 19 | \( 1 + 4.48T + 19T^{2} \) |
| 23 | \( 1 - 4.40T + 23T^{2} \) |
| 29 | \( 1 - 2.36T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 - 5.60T + 37T^{2} \) |
| 41 | \( 1 - 4.45T + 41T^{2} \) |
| 43 | \( 1 + 8.18T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 7.06T + 59T^{2} \) |
| 61 | \( 1 - 3.58T + 61T^{2} \) |
| 67 | \( 1 - 1.70T + 67T^{2} \) |
| 71 | \( 1 - 5.36T + 71T^{2} \) |
| 73 | \( 1 + 0.516T + 73T^{2} \) |
| 79 | \( 1 + 6.93T + 79T^{2} \) |
| 83 | \( 1 - 8.45T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.77017535599610639668249064672, −6.98527774205985799206233459249, −6.51101974206982061735272284738, −5.41265698118216387642453782787, −4.50300111064694421492322061547, −4.07014941532908230250000988857, −2.77792929552206140440468309670, −2.01407319119254822915919837170, −0.991677929882646542807892364981, 0,
0.991677929882646542807892364981, 2.01407319119254822915919837170, 2.77792929552206140440468309670, 4.07014941532908230250000988857, 4.50300111064694421492322061547, 5.41265698118216387642453782787, 6.51101974206982061735272284738, 6.98527774205985799206233459249, 7.77017535599610639668249064672
