| L(s) = 1 | + 2-s + 4-s + 4·5-s − 4·7-s + 8-s + 4·10-s + 5·11-s − 6·13-s − 4·14-s + 16-s − 17-s + 4·20-s + 5·22-s + 5·23-s + 11·25-s − 6·26-s − 4·28-s + 3·29-s + 2·31-s + 32-s − 34-s − 16·35-s + 5·37-s + 4·40-s + 5·41-s − 6·43-s + 5·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s − 1.51·7-s + 0.353·8-s + 1.26·10-s + 1.50·11-s − 1.66·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.894·20-s + 1.06·22-s + 1.04·23-s + 11/5·25-s − 1.17·26-s − 0.755·28-s + 0.557·29-s + 0.359·31-s + 0.176·32-s − 0.171·34-s − 2.70·35-s + 0.821·37-s + 0.632·40-s + 0.780·41-s − 0.914·43-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.881193620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.881193620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.741064026437871071398327882661, −7.24986704396457291491313125350, −6.64535823136021067696942954803, −6.35659472166838845634375802997, −5.52042470002543942564588374092, −4.81975683265323839193934675019, −3.82107607914526795119779242713, −2.77892436303685197747246750529, −2.32822660208939823909793939198, −1.05245083617504106324557102864,
1.05245083617504106324557102864, 2.32822660208939823909793939198, 2.77892436303685197747246750529, 3.82107607914526795119779242713, 4.81975683265323839193934675019, 5.52042470002543942564588374092, 6.35659472166838845634375802997, 6.64535823136021067696942954803, 7.24986704396457291491313125350, 8.741064026437871071398327882661
