| L(s) = 1 | + 2.68·2-s + 5.21·4-s + 3.72·5-s + 7-s + 8.62·8-s + 10.0·10-s − 4.70·11-s − 1.94·13-s + 2.68·14-s + 12.7·16-s + 6.80·17-s − 1.46·19-s + 19.4·20-s − 12.6·22-s + 1.68·23-s + 8.87·25-s − 5.21·26-s + 5.21·28-s − 5.63·29-s + 8.82·31-s + 16.9·32-s + 18.2·34-s + 3.72·35-s − 10.5·37-s − 3.93·38-s + 32.1·40-s + 10.3·41-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 2.60·4-s + 1.66·5-s + 0.377·7-s + 3.04·8-s + 3.16·10-s − 1.41·11-s − 0.539·13-s + 0.717·14-s + 3.18·16-s + 1.64·17-s − 0.335·19-s + 4.34·20-s − 2.69·22-s + 0.350·23-s + 1.77·25-s − 1.02·26-s + 0.985·28-s − 1.04·29-s + 1.58·31-s + 2.99·32-s + 3.13·34-s + 0.629·35-s − 1.73·37-s − 0.637·38-s + 5.08·40-s + 1.61·41-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
| \(L(1)\) |
\(\approx\) |
\(10.60614894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.60614894\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.68T + 2T^{2} \) |
| 5 | \( 1 - 3.72T + 5T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 - 6.80T + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 - 1.68T + 23T^{2} \) |
| 29 | \( 1 + 5.63T + 29T^{2} \) |
| 31 | \( 1 - 8.82T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4.90T + 43T^{2} \) |
| 47 | \( 1 - 1.59T + 47T^{2} \) |
| 53 | \( 1 + 3.28T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 + 7.92T + 61T^{2} \) |
| 67 | \( 1 - 6.80T + 67T^{2} \) |
| 71 | \( 1 - 8.27T + 71T^{2} \) |
| 73 | \( 1 + 3.44T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 0.628T + 83T^{2} \) |
| 89 | \( 1 - 5.11T + 89T^{2} \) |
| 97 | \( 1 + 8.17T + 97T^{2} \) |
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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
−7.59464023272100693270361890613, −6.83548967772199688164049656795, −6.06627729665049401627839839291, −5.57654853775315875583959397999, −5.12392569014438118375751304644, −4.62733551170800689109383659911, −3.42043046246076908954213801440, −2.76389729788360779090027037596, −2.17645393714558269569184194345, −1.37642073269460811546110354838,
1.37642073269460811546110354838, 2.17645393714558269569184194345, 2.76389729788360779090027037596, 3.42043046246076908954213801440, 4.62733551170800689109383659911, 5.12392569014438118375751304644, 5.57654853775315875583959397999, 6.06627729665049401627839839291, 6.83548967772199688164049656795, 7.59464023272100693270361890613
