| L(s) = 1 | + 3-s − 4.23·5-s + 2.23·7-s − 2·9-s + 0.763·11-s + 1.23·13-s − 4.23·15-s − 17-s − 5.47·19-s + 2.23·21-s + 4.76·23-s + 12.9·25-s − 5·27-s + 2.70·29-s + 9.23·31-s + 0.763·33-s − 9.47·35-s − 8·37-s + 1.23·39-s + 0.527·41-s − 5.23·43-s + 8.47·45-s + 10.4·47-s − 1.99·49-s − 51-s + 0.708·53-s − 3.23·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.89·5-s + 0.845·7-s − 0.666·9-s + 0.230·11-s + 0.342·13-s − 1.09·15-s − 0.242·17-s − 1.25·19-s + 0.487·21-s + 0.993·23-s + 2.58·25-s − 0.962·27-s + 0.502·29-s + 1.65·31-s + 0.132·33-s − 1.60·35-s − 1.31·37-s + 0.197·39-s + 0.0824·41-s − 0.798·43-s + 1.26·45-s + 1.52·47-s − 0.285·49-s − 0.140·51-s + 0.0972·53-s − 0.436·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 + 4.23T + 5T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 19 | \( 1 + 5.47T + 19T^{2} \) |
| 23 | \( 1 - 4.76T + 23T^{2} \) |
| 29 | \( 1 - 2.70T + 29T^{2} \) |
| 31 | \( 1 - 9.23T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 0.527T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 0.708T + 53T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 3.70T + 73T^{2} \) |
| 79 | \( 1 - 8.23T + 79T^{2} \) |
| 83 | \( 1 - 3.23T + 83T^{2} \) |
| 89 | \( 1 - 6.94T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.75938509457618106660737043157, −6.92943730020841640239599032291, −6.30989888488252808969707571308, −5.09655536517777774478389108158, −4.54882287144890496656361233221, −3.88233178260442395117308380610, −3.20142420370223324243790678197, −2.42663614805620879654685015597, −1.15450175864054094946173479721, 0,
1.15450175864054094946173479721, 2.42663614805620879654685015597, 3.20142420370223324243790678197, 3.88233178260442395117308380610, 4.54882287144890496656361233221, 5.09655536517777774478389108158, 6.30989888488252808969707571308, 6.92943730020841640239599032291, 7.75938509457618106660737043157
