| L(s) = 1 | + 1.33·2-s − 0.227·4-s + 2.92·5-s − 7-s − 2.96·8-s + 3.89·10-s − 6.16·11-s − 0.620·13-s − 1.33·14-s − 3.49·16-s − 0.644·17-s + 2.75·19-s − 0.665·20-s − 8.21·22-s − 4.71·23-s + 3.55·25-s − 0.825·26-s + 0.227·28-s + 8.85·29-s + 2.41·31-s + 1.28·32-s − 0.858·34-s − 2.92·35-s − 6.07·37-s + 3.66·38-s − 8.67·40-s + 2.70·41-s + ⋯ |
| L(s) = 1 | + 0.941·2-s − 0.113·4-s + 1.30·5-s − 0.377·7-s − 1.04·8-s + 1.23·10-s − 1.86·11-s − 0.171·13-s − 0.355·14-s − 0.873·16-s − 0.156·17-s + 0.632·19-s − 0.148·20-s − 1.75·22-s − 0.982·23-s + 0.710·25-s − 0.161·26-s + 0.0429·28-s + 1.64·29-s + 0.434·31-s + 0.226·32-s − 0.147·34-s − 0.494·35-s − 0.998·37-s + 0.595·38-s − 1.37·40-s + 0.421·41-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
| \(L(1)\) |
\(\approx\) |
\(2.712479782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.712479782\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.33T + 2T^{2} \) |
| 5 | \( 1 - 2.92T + 5T^{2} \) |
| 11 | \( 1 + 6.16T + 11T^{2} \) |
| 13 | \( 1 + 0.620T + 13T^{2} \) |
| 17 | \( 1 + 0.644T + 17T^{2} \) |
| 19 | \( 1 - 2.75T + 19T^{2} \) |
| 23 | \( 1 + 4.71T + 23T^{2} \) |
| 29 | \( 1 - 8.85T + 29T^{2} \) |
| 31 | \( 1 - 2.41T + 31T^{2} \) |
| 37 | \( 1 + 6.07T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 6.17T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 8.20T + 73T^{2} \) |
| 79 | \( 1 - 2.30T + 79T^{2} \) |
| 83 | \( 1 - 1.22T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 1.19T + 97T^{2} \) |
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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
−7.83231981299040171242205719963, −6.91345813675832136317254987265, −6.10708455047230977392366444038, −5.67262500369249470742752609182, −5.12099865087015664199990794704, −4.49616856630472966916171664236, −3.48534800646073288522669260597, −2.61343053539112679880103849686, −2.27717682439996607026152289163, −0.67570269412527924252378746248,
0.67570269412527924252378746248, 2.27717682439996607026152289163, 2.61343053539112679880103849686, 3.48534800646073288522669260597, 4.49616856630472966916171664236, 5.12099865087015664199990794704, 5.67262500369249470742752609182, 6.10708455047230977392366444038, 6.91345813675832136317254987265, 7.83231981299040171242205719963
