| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 2·11-s + 4·13-s − 15-s + 2·17-s + 2·19-s − 21-s + 4·23-s + 25-s − 27-s + 6·29-s − 2·31-s + 2·33-s + 35-s + 10·37-s − 4·39-s − 10·41-s + 12·43-s + 45-s − 8·47-s + 49-s − 2·51-s − 2·55-s − 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s − 0.258·15-s + 0.485·17-s + 0.458·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.348·33-s + 0.169·35-s + 1.64·37-s − 0.640·39-s − 1.56·41-s + 1.82·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.339758166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.339758166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−11.09779642116460041629796988622, −10.46678414017466153558768634372, −9.484041131040810573980520780565, −8.444776128815454998541535245368, −7.46985318735628348223218545659, −6.31888690467862628757406619126, −5.51088025071477982042675051210, −4.51008988119235510509832391491, −3.00191084542596222591694033423, −1.28136430783138726696385588581,
1.28136430783138726696385588581, 3.00191084542596222591694033423, 4.51008988119235510509832391491, 5.51088025071477982042675051210, 6.31888690467862628757406619126, 7.46985318735628348223218545659, 8.444776128815454998541535245368, 9.484041131040810573980520780565, 10.46678414017466153558768634372, 11.09779642116460041629796988622
