L(s) = 1 | + 1.86·2-s + 1.48·4-s + 4.06·5-s − 7-s − 0.960·8-s + 7.59·10-s − 0.0552·11-s − 5.30·13-s − 1.86·14-s − 4.76·16-s + 1.80·17-s + 7.67·19-s + 6.04·20-s − 0.103·22-s + 8.46·23-s + 11.5·25-s − 9.90·26-s − 1.48·28-s + 2.19·29-s + 7.83·31-s − 6.97·32-s + 3.37·34-s − 4.06·35-s + 2.03·37-s + 14.3·38-s − 3.90·40-s − 2.50·41-s + ⋯ |
L(s) = 1 | + 1.32·2-s + 0.742·4-s + 1.82·5-s − 0.377·7-s − 0.339·8-s + 2.40·10-s − 0.0166·11-s − 1.47·13-s − 0.498·14-s − 1.19·16-s + 0.438·17-s + 1.76·19-s + 1.35·20-s − 0.0220·22-s + 1.76·23-s + 2.31·25-s − 1.94·26-s − 0.280·28-s + 0.406·29-s + 1.40·31-s − 1.23·32-s + 0.578·34-s − 0.687·35-s + 0.334·37-s + 2.32·38-s − 0.617·40-s − 0.391·41-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
\(L(1)\) |
\(\approx\) |
\(5.688621928\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.688621928\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.86T + 2T^{2} \) |
| 5 | \( 1 - 4.06T + 5T^{2} \) |
| 11 | \( 1 + 0.0552T + 11T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 17 | \( 1 - 1.80T + 17T^{2} \) |
| 19 | \( 1 - 7.67T + 19T^{2} \) |
| 23 | \( 1 - 8.46T + 23T^{2} \) |
| 29 | \( 1 - 2.19T + 29T^{2} \) |
| 31 | \( 1 - 7.83T + 31T^{2} \) |
| 37 | \( 1 - 2.03T + 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 + 9.74T + 43T^{2} \) |
| 47 | \( 1 + 5.74T + 47T^{2} \) |
| 53 | \( 1 + 3.80T + 53T^{2} \) |
| 59 | \( 1 + 7.25T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 1.02T + 67T^{2} \) |
| 71 | \( 1 - 4.59T + 71T^{2} \) |
| 73 | \( 1 + 0.190T + 73T^{2} \) |
| 79 | \( 1 - 8.89T + 79T^{2} \) |
| 83 | \( 1 + 4.28T + 83T^{2} \) |
| 89 | \( 1 + 8.76T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.51728643061059092732708944906, −6.72809497402230766457449513484, −6.38810871170669007409626939825, −5.44259739682497427719162587086, −5.11204694604814787130869023365, −4.69501842696593792961509217662, −3.19704250753202982282614566031, −2.97489721664727127399550517633, −2.13165768668374388688190782431, −0.993093473973767803921559251980,
0.993093473973767803921559251980, 2.13165768668374388688190782431, 2.97489721664727127399550517633, 3.19704250753202982282614566031, 4.69501842696593792961509217662, 5.11204694604814787130869023365, 5.44259739682497427719162587086, 6.38810871170669007409626939825, 6.72809497402230766457449513484, 7.51728643061059092732708944906