| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 5·11-s − 2·13-s + 14-s + 16-s + 4·17-s + 2·19-s + 20-s + 5·22-s − 4·25-s − 2·26-s + 28-s − 2·29-s + 3·31-s + 32-s + 4·34-s + 35-s − 8·37-s + 2·38-s + 40-s − 4·43-s + 5·44-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 + 1.50·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.458·19-s + 0.223·20-s + 1.06·22-s − 4/5·25-s − 0.392·26-s + 0.188·28-s − 0.371·29-s + 0.538·31-s + 0.176·32-s + 0.685·34-s + 0.169·35-s − 1.31·37-s + 0.324·38-s + 0.158·40-s − 0.609·43-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.295818841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.295818841\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−14.96554761225169, −14.52709446494531, −14.27641541810838, −13.62304757710173, −13.24007210546088, −12.44041652975777, −11.93836891222703, −11.67882247348104, −11.12956778219152, −10.15148062396323, −9.958250808784286, −9.318355938316154, −8.607810331260243, −8.028572316621927, −7.280711584691651, −6.823598670402345, −6.220411216407207, −5.530837995499325, −5.127175383677410, −4.371271474942457, −3.667989937723049, −3.247510318666765, −2.174437548306609, −1.670048692591826, −0.8197502218700847,
0.8197502218700847, 1.670048692591826, 2.174437548306609, 3.247510318666765, 3.667989937723049, 4.371271474942457, 5.127175383677410, 5.530837995499325, 6.220411216407207, 6.823598670402345, 7.280711584691651, 8.028572316621927, 8.607810331260243, 9.318355938316154, 9.958250808784286, 10.15148062396323, 11.12956778219152, 11.67882247348104, 11.93836891222703, 12.44041652975777, 13.24007210546088, 13.62304757710173, 14.27641541810838, 14.52709446494531, 14.96554761225169
