| L(s) = 1 | + 1.76·3-s + 0.187·5-s − 7-s + 0.124·9-s − 11-s − 13-s + 0.330·15-s + 4.32·17-s − 5.79·19-s − 1.76·21-s + 5.19·23-s − 4.96·25-s − 5.08·27-s + 1.25·29-s + 6.52·31-s − 1.76·33-s − 0.187·35-s − 5.96·37-s − 1.76·39-s + 5.23·41-s − 3.99·43-s + 0.0233·45-s − 5.21·47-s + 49-s + 7.64·51-s − 1.33·53-s − 0.187·55-s + ⋯ |
| L(s) = 1 | + 1.02·3-s + 0.0837·5-s − 0.377·7-s + 0.0415·9-s − 0.301·11-s − 0.277·13-s + 0.0854·15-s + 1.04·17-s − 1.33·19-s − 0.385·21-s + 1.08·23-s − 0.992·25-s − 0.978·27-s + 0.233·29-s + 1.17·31-s − 0.307·33-s − 0.0316·35-s − 0.981·37-s − 0.283·39-s + 0.817·41-s − 0.609·43-s + 0.00347·45-s − 0.761·47-s + 0.142·49-s + 1.07·51-s − 0.183·53-s − 0.0252·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 - 0.187T + 5T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 + 5.79T + 19T^{2} \) |
| 23 | \( 1 - 5.19T + 23T^{2} \) |
| 29 | \( 1 - 1.25T + 29T^{2} \) |
| 31 | \( 1 - 6.52T + 31T^{2} \) |
| 37 | \( 1 + 5.96T + 37T^{2} \) |
| 41 | \( 1 - 5.23T + 41T^{2} \) |
| 43 | \( 1 + 3.99T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 + 1.33T + 53T^{2} \) |
| 59 | \( 1 - 7.28T + 59T^{2} \) |
| 61 | \( 1 + 3.40T + 61T^{2} \) |
| 67 | \( 1 + 2.43T + 67T^{2} \) |
| 71 | \( 1 + 2.65T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 3.58T + 79T^{2} \) |
| 83 | \( 1 + 9.04T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 4.31T + 97T^{2} \) |
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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
−7.60898071764560603237538443679, −6.91560788424068990734840695246, −6.12086010589713446218770441441, −5.43743811075816633216713027236, −4.55176307067779735222353947488, −3.73797860725752906589876368468, −2.99372050873294885968319230066, −2.45894774798311388434405949580, −1.44623589490272966392194170247, 0,
1.44623589490272966392194170247, 2.45894774798311388434405949580, 2.99372050873294885968319230066, 3.73797860725752906589876368468, 4.55176307067779735222353947488, 5.43743811075816633216713027236, 6.12086010589713446218770441441, 6.91560788424068990734840695246, 7.60898071764560603237538443679
