| L(s) = 1 | + 0.470·3-s − 2.71·7-s − 2.77·9-s − 5.55·11-s − 2.02·13-s + 3.77·17-s + 19-s − 1.28·21-s + 5.77·23-s − 2.71·27-s − 5.66·29-s + 7.55·31-s − 2.61·33-s + 3.75·37-s − 0.954·39-s − 12.6·41-s + 9.43·43-s − 11.1·47-s + 0.397·49-s + 1.77·51-s + 8.85·53-s + 0.470·57-s + 11.4·59-s − 10.6·61-s + 7.55·63-s + 11.5·67-s + 2.71·69-s + ⋯ |
| L(s) = 1 | + 0.271·3-s − 1.02·7-s − 0.926·9-s − 1.67·11-s − 0.562·13-s + 0.916·17-s + 0.229·19-s − 0.279·21-s + 1.20·23-s − 0.523·27-s − 1.05·29-s + 1.35·31-s − 0.455·33-s + 0.616·37-s − 0.152·39-s − 1.97·41-s + 1.43·43-s − 1.62·47-s + 0.0567·49-s + 0.249·51-s + 1.21·53-s + 0.0623·57-s + 1.49·59-s − 1.35·61-s + 0.952·63-s + 1.41·67-s + 0.327·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.106878961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.106878961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.470T + 3T^{2} \) |
| 7 | \( 1 + 2.71T + 7T^{2} \) |
| 11 | \( 1 + 5.55T + 11T^{2} \) |
| 13 | \( 1 + 2.02T + 13T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 23 | \( 1 - 5.77T + 23T^{2} \) |
| 29 | \( 1 + 5.66T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 - 3.75T + 37T^{2} \) |
| 41 | \( 1 + 12.6T + 41T^{2} \) |
| 43 | \( 1 - 9.43T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 8.85T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 9.45T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.351274907500117044953471704615, −7.87775982919645798907947903913, −7.09579339567574527195235920685, −6.26398519236206470859068217549, −5.38408215143746150094958780856, −4.98937258292318795187056659972, −3.54672106772336105030415060789, −2.97825810075499499302730386834, −2.32016214062698605115597524846, −0.56449189064198848543220873619,
0.56449189064198848543220873619, 2.32016214062698605115597524846, 2.97825810075499499302730386834, 3.54672106772336105030415060789, 4.98937258292318795187056659972, 5.38408215143746150094958780856, 6.26398519236206470859068217549, 7.09579339567574527195235920685, 7.87775982919645798907947903913, 8.351274907500117044953471704615
