| L(s) = 1 | + 3-s − 5-s − 7-s − 2·9-s + 4·11-s − 4·13-s − 15-s + 4·17-s − 19-s − 21-s + 4·23-s − 4·25-s − 5·27-s − 3·29-s + 2·31-s + 4·33-s + 35-s + 8·37-s − 4·39-s − 2·41-s − 8·43-s + 2·45-s + 47-s + 49-s + 4·51-s + 8·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 1.20·11-s − 1.10·13-s − 0.258·15-s + 0.970·17-s − 0.229·19-s − 0.218·21-s + 0.834·23-s − 4/5·25-s − 0.962·27-s − 0.557·29-s + 0.359·31-s + 0.696·33-s + 0.169·35-s + 1.31·37-s − 0.640·39-s − 0.312·41-s − 1.21·43-s + 0.298·45-s + 0.145·47-s + 1/7·49-s + 0.560·51-s + 1.09·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.876543286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.876543286\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.65110592609255, −14.16289271714615, −13.43492282460247, −13.17335255306155, −12.25286769937598, −12.05620526320461, −11.49804422933183, −11.09557036357951, −10.19386576343669, −9.712995081746478, −9.397876112829592, −8.711264263136875, −8.305841458895434, −7.613986489400372, −7.253141015273110, −6.567036895421626, −5.982099622156157, −5.353648869340106, −4.719796605532377, −3.860678955621856, −3.606836497157793, −2.819918873662503, −2.304914268369030, −1.379941861956824, −0.4722083573458134,
0.4722083573458134, 1.379941861956824, 2.304914268369030, 2.819918873662503, 3.606836497157793, 3.860678955621856, 4.719796605532377, 5.353648869340106, 5.982099622156157, 6.567036895421626, 7.253141015273110, 7.613986489400372, 8.305841458895434, 8.711264263136875, 9.397876112829592, 9.712995081746478, 10.19386576343669, 11.09557036357951, 11.49804422933183, 12.05620526320461, 12.25286769937598, 13.17335255306155, 13.43492282460247, 14.16289271714615, 14.65110592609255
