| L(s) = 1 | + 0.358·3-s + 5-s + 1.78·7-s − 2.87·9-s + 3.64·11-s − 1.14·13-s + 0.358·15-s + 4.34·17-s − 6.49·19-s + 0.639·21-s + 4.28·23-s + 25-s − 2.10·27-s − 8.72·29-s − 10.1·31-s + 1.30·33-s + 1.78·35-s − 5.97·37-s − 0.410·39-s + 1.06·41-s − 8.89·43-s − 2.87·45-s − 2.51·47-s − 3.81·49-s + 1.55·51-s − 3.49·53-s + 3.64·55-s + ⋯ |
| L(s) = 1 | + 0.206·3-s + 0.447·5-s + 0.674·7-s − 0.957·9-s + 1.10·11-s − 0.318·13-s + 0.0924·15-s + 1.05·17-s − 1.48·19-s + 0.139·21-s + 0.894·23-s + 0.200·25-s − 0.404·27-s − 1.61·29-s − 1.81·31-s + 0.227·33-s + 0.301·35-s − 0.982·37-s − 0.0657·39-s + 0.166·41-s − 1.35·43-s − 0.428·45-s − 0.366·47-s − 0.544·49-s + 0.217·51-s − 0.480·53-s + 0.492·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 0.358T + 3T^{2} \) |
| 7 | \( 1 - 1.78T + 7T^{2} \) |
| 11 | \( 1 - 3.64T + 11T^{2} \) |
| 13 | \( 1 + 1.14T + 13T^{2} \) |
| 17 | \( 1 - 4.34T + 17T^{2} \) |
| 19 | \( 1 + 6.49T + 19T^{2} \) |
| 23 | \( 1 - 4.28T + 23T^{2} \) |
| 29 | \( 1 + 8.72T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 5.97T + 37T^{2} \) |
| 41 | \( 1 - 1.06T + 41T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 + 2.51T + 47T^{2} \) |
| 53 | \( 1 + 3.49T + 53T^{2} \) |
| 59 | \( 1 - 1.19T + 59T^{2} \) |
| 61 | \( 1 + 9.61T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 0.472T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 4.75T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 3.41T + 97T^{2} \) |
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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
−7.49385959710367764421285916924, −6.82309321133881173256446760885, −6.02390508470861447140392802586, −5.43278325234962683983727444903, −4.77586345501347759991862452156, −3.75785544980621454078838727155, −3.20310658461457526113973843851, −2.03433037567835997789730176361, −1.52935112617448434593268877402, 0,
1.52935112617448434593268877402, 2.03433037567835997789730176361, 3.20310658461457526113973843851, 3.75785544980621454078838727155, 4.77586345501347759991862452156, 5.43278325234962683983727444903, 6.02390508470861447140392802586, 6.82309321133881173256446760885, 7.49385959710367764421285916924
