| L(s) = 1 | − 3-s + 1.42·5-s − 1.49·7-s + 9-s + 4.45·11-s − 0.159·13-s − 1.42·15-s + 3.47·17-s + 1.29·19-s + 1.49·21-s + 23-s − 2.96·25-s − 27-s − 29-s − 8.87·31-s − 4.45·33-s − 2.12·35-s − 9.07·37-s + 0.159·39-s − 6.40·41-s − 12.0·43-s + 1.42·45-s − 10.2·47-s − 4.77·49-s − 3.47·51-s + 8.00·53-s + 6.35·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.637·5-s − 0.563·7-s + 0.333·9-s + 1.34·11-s − 0.0440·13-s − 0.368·15-s + 0.842·17-s + 0.297·19-s + 0.325·21-s + 0.208·23-s − 0.593·25-s − 0.192·27-s − 0.185·29-s − 1.59·31-s − 0.776·33-s − 0.359·35-s − 1.49·37-s + 0.0254·39-s − 1.00·41-s − 1.83·43-s + 0.212·45-s − 1.49·47-s − 0.682·49-s − 0.486·51-s + 1.09·53-s + 0.857·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 1.42T + 5T^{2} \) |
| 7 | \( 1 + 1.49T + 7T^{2} \) |
| 11 | \( 1 - 4.45T + 11T^{2} \) |
| 13 | \( 1 + 0.159T + 13T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 19 | \( 1 - 1.29T + 19T^{2} \) |
| 31 | \( 1 + 8.87T + 31T^{2} \) |
| 37 | \( 1 + 9.07T + 37T^{2} \) |
| 41 | \( 1 + 6.40T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 8.00T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 0.485T + 67T^{2} \) |
| 71 | \( 1 - 7.64T + 71T^{2} \) |
| 73 | \( 1 + 3.71T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 1.30T + 83T^{2} \) |
| 89 | \( 1 + 18.2T + 89T^{2} \) |
| 97 | \( 1 - 2.78T + 97T^{2} \) |
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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
−7.19770457294778528086093844358, −6.75999072415147620276881481363, −6.13126693306485412386717421128, −5.43789878113671765188587451018, −4.89358811047497625707742084228, −3.63435970841590504283821934425, −3.44177881151764552085067872042, −1.95649843065576687548074786143, −1.35485314070441286542663322758, 0,
1.35485314070441286542663322758, 1.95649843065576687548074786143, 3.44177881151764552085067872042, 3.63435970841590504283821934425, 4.89358811047497625707742084228, 5.43789878113671765188587451018, 6.13126693306485412386717421128, 6.75999072415147620276881481363, 7.19770457294778528086093844358
