| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 3·9-s − 6·11-s − 5·13-s + 14-s + 16-s + 5·17-s + 3·18-s + 4·19-s + 6·22-s + 9·23-s + 5·26-s − 28-s − 3·29-s − 32-s − 5·34-s − 3·36-s + 8·37-s − 4·38-s + 4·43-s − 6·44-s − 9·46-s − 7·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 9-s − 1.80·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s + 1.21·17-s + 0.707·18-s + 0.917·19-s + 1.27·22-s + 1.87·23-s + 0.980·26-s − 0.188·28-s − 0.557·29-s − 0.176·32-s − 0.857·34-s − 1/2·36-s + 1.31·37-s − 0.648·38-s + 0.609·43-s − 0.904·44-s − 1.32·46-s − 1.02·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.212940812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.212940812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 293 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.67946778259965, −13.05673272301710, −12.85981820709806, −12.12233732124542, −11.82445522966710, −11.03954900670951, −10.88511248075278, −10.18768460213465, −9.683688068574749, −9.458661482115675, −8.801550916545592, −8.133056822583795, −7.702565172278912, −7.452093920436301, −6.853989584795816, −6.070390935054229, −5.527242444705913, −5.099015975076094, −4.773315811846099, −3.384433656866377, −3.224305530895792, −2.500873226900685, −2.209757084134897, −0.9044411922701802, −0.4859790057330606,
0.4859790057330606, 0.9044411922701802, 2.209757084134897, 2.500873226900685, 3.224305530895792, 3.384433656866377, 4.773315811846099, 5.099015975076094, 5.527242444705913, 6.070390935054229, 6.853989584795816, 7.452093920436301, 7.702565172278912, 8.133056822583795, 8.801550916545592, 9.458661482115675, 9.683688068574749, 10.18768460213465, 10.88511248075278, 11.03954900670951, 11.82445522966710, 12.12233732124542, 12.85981820709806, 13.05673272301710, 13.67946778259965
