| L(s) = 1 | − 2.14·3-s + 2.05·7-s + 1.60·9-s − 0.879·11-s − 2.89·13-s + 4.30·17-s + 6.14·19-s − 4.41·21-s + 4.27·23-s + 3.00·27-s + 2.60·29-s + 0.366·31-s + 1.88·33-s + 10.5·37-s + 6.20·39-s − 2.84·41-s + 7.77·43-s − 7.33·47-s − 2.77·49-s − 9.23·51-s − 9.25·53-s − 13.1·57-s + 11.0·59-s − 8.23·61-s + 3.29·63-s − 0.282·67-s − 9.16·69-s + ⋯ |
| L(s) = 1 | − 1.23·3-s + 0.777·7-s + 0.533·9-s − 0.265·11-s − 0.802·13-s + 1.04·17-s + 1.40·19-s − 0.962·21-s + 0.890·23-s + 0.577·27-s + 0.484·29-s + 0.0657·31-s + 0.328·33-s + 1.73·37-s + 0.993·39-s − 0.444·41-s + 1.18·43-s − 1.06·47-s − 0.396·49-s − 1.29·51-s − 1.27·53-s − 1.74·57-s + 1.43·59-s − 1.05·61-s + 0.414·63-s − 0.0345·67-s − 1.10·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.464488377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.464488377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.14T + 3T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 11 | \( 1 + 0.879T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 19 | \( 1 - 6.14T + 19T^{2} \) |
| 23 | \( 1 - 4.27T + 23T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 - 0.366T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 2.84T + 41T^{2} \) |
| 43 | \( 1 - 7.77T + 43T^{2} \) |
| 47 | \( 1 + 7.33T + 47T^{2} \) |
| 53 | \( 1 + 9.25T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 8.23T + 61T^{2} \) |
| 67 | \( 1 + 0.282T + 67T^{2} \) |
| 71 | \( 1 - 2.50T + 71T^{2} \) |
| 73 | \( 1 + 1.11T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 - 6.22T + 97T^{2} \) |
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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
−7.61842540234411559630244257391, −6.98860251200881604807625600847, −6.12048073806600424975453858543, −5.57613736107832254369492870527, −4.89341709958944091982635580311, −4.64975491871298137420958197330, −3.35596025796961165405086325644, −2.63987713011631601407083108305, −1.38819242870841295739504204774, −0.67721502038463588134363631258,
0.67721502038463588134363631258, 1.38819242870841295739504204774, 2.63987713011631601407083108305, 3.35596025796961165405086325644, 4.64975491871298137420958197330, 4.89341709958944091982635580311, 5.57613736107832254369492870527, 6.12048073806600424975453858543, 6.98860251200881604807625600847, 7.61842540234411559630244257391
