| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 11-s − 13-s + 14-s + 16-s + 3·17-s + 19-s − 20-s + 22-s + 2·23-s + 25-s + 26-s − 28-s − 29-s − 3·31-s − 32-s − 3·34-s + 35-s + 7·37-s − 38-s + 40-s + 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.229·19-s − 0.223·20-s + 0.213·22-s + 0.417·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s − 0.185·29-s − 0.538·31-s − 0.176·32-s − 0.514·34-s + 0.169·35-s + 1.15·37-s − 0.162·38-s + 0.158·40-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9959874503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9959874503\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−16.32698155817147, −15.85620676221974, −15.25708941536639, −14.63873735219243, −14.24506526556146, −13.27707860507553, −12.79948546565327, −12.29860631086810, −11.53385211138605, −11.15597422498113, −10.49025940782162, −9.750865025017108, −9.490794845029796, −8.681570091659570, −8.021389879691198, −7.573625353933547, −6.961383533925640, −6.252918715809611, −5.547389897196516, −4.827426077347919, −3.925432576898583, −3.173172033082770, −2.537463406139926, −1.474746836017295, −0.5166429165382640,
0.5166429165382640, 1.474746836017295, 2.537463406139926, 3.173172033082770, 3.925432576898583, 4.827426077347919, 5.547389897196516, 6.252918715809611, 6.961383533925640, 7.573625353933547, 8.021389879691198, 8.681570091659570, 9.490794845029796, 9.750865025017108, 10.49025940782162, 11.15597422498113, 11.53385211138605, 12.29860631086810, 12.79948546565327, 13.27707860507553, 14.24506526556146, 14.63873735219243, 15.25708941536639, 15.85620676221974, 16.32698155817147
