| L(s) = 1 | + 2.48·2-s + 4.19·4-s − 1.44·5-s − 7-s + 5.47·8-s − 3.60·10-s + 3.27·11-s − 3.24·13-s − 2.48·14-s + 5.23·16-s − 2.00·17-s − 7.49·19-s − 6.08·20-s + 8.15·22-s + 5.11·23-s − 2.90·25-s − 8.08·26-s − 4.19·28-s + 2.27·29-s − 9.36·31-s + 2.08·32-s − 4.98·34-s + 1.44·35-s + 6.96·37-s − 18.6·38-s − 7.93·40-s − 11.0·41-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 2.09·4-s − 0.647·5-s − 0.377·7-s + 1.93·8-s − 1.14·10-s + 0.987·11-s − 0.901·13-s − 0.665·14-s + 1.30·16-s − 0.485·17-s − 1.71·19-s − 1.35·20-s + 1.73·22-s + 1.06·23-s − 0.580·25-s − 1.58·26-s − 0.793·28-s + 0.421·29-s − 1.68·31-s + 0.368·32-s − 0.854·34-s + 0.244·35-s + 1.14·37-s − 3.02·38-s − 1.25·40-s − 1.72·41-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 | \( 1 - 2.48T + 2T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 + 3.24T + 13T^{2} \) |
| 17 | \( 1 + 2.00T + 17T^{2} \) |
| 19 | \( 1 + 7.49T + 19T^{2} \) |
| 23 | \( 1 - 5.11T + 23T^{2} \) |
| 29 | \( 1 - 2.27T + 29T^{2} \) |
| 31 | \( 1 + 9.36T + 31T^{2} \) |
| 37 | \( 1 - 6.96T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 7.34T + 43T^{2} \) |
| 47 | \( 1 + 3.00T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 8.25T + 59T^{2} \) |
| 61 | \( 1 - 9.29T + 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 + 6.76T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 6.43T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 9.26T + 89T^{2} \) |
| 97 | \( 1 - 3.36T + 97T^{2} \) |
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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
−6.96333907963454091153272053935, −6.76914392874094003933679677400, −6.08776590565069923902335684569, −5.15333642487962944991039392509, −4.62671685081520970585490830248, −3.90525850200667422282806956161, −3.46312522783334464503619288418, −2.51058290608396911246773116682, −1.74460808437066896212049488996, 0,
1.74460808437066896212049488996, 2.51058290608396911246773116682, 3.46312522783334464503619288418, 3.90525850200667422282806956161, 4.62671685081520970585490830248, 5.15333642487962944991039392509, 6.08776590565069923902335684569, 6.76914392874094003933679677400, 6.96333907963454091153272053935
