| L(s) = 1 | − 3-s + 9-s + 4·11-s + 13-s − 5·17-s − 6·19-s + 5·23-s − 27-s − 3·29-s − 7·31-s − 4·33-s + 4·37-s − 39-s − 5·41-s + 43-s − 8·47-s + 5·51-s + 5·53-s + 6·57-s − 59-s − 61-s − 4·67-s − 5·69-s + 12·71-s − 4·73-s + 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 1.21·17-s − 1.37·19-s + 1.04·23-s − 0.192·27-s − 0.557·29-s − 1.25·31-s − 0.696·33-s + 0.657·37-s − 0.160·39-s − 0.780·41-s + 0.152·43-s − 1.16·47-s + 0.700·51-s + 0.686·53-s + 0.794·57-s − 0.130·59-s − 0.128·61-s − 0.488·67-s − 0.601·69-s + 1.42·71-s − 0.468·73-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| 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
−14.69525566462333, −14.12348984511172, −13.41308044222175, −13.02568053235449, −12.65441188437685, −11.99437313111815, −11.43727442082865, −11.03979080178682, −10.74655470150871, −10.00994953494050, −9.348017938032347, −8.941666527059418, −8.564753049517119, −7.765734072959891, −7.103508961610372, −6.545686187460054, −6.366625732814313, −5.608908243580227, −4.917512736268140, −4.406633885002424, −3.838048687288707, −3.277476907116321, −2.197464553460955, −1.773254492979313, −0.8738377866025912, 0,
0.8738377866025912, 1.773254492979313, 2.197464553460955, 3.277476907116321, 3.838048687288707, 4.406633885002424, 4.917512736268140, 5.608908243580227, 6.366625732814313, 6.545686187460054, 7.103508961610372, 7.765734072959891, 8.564753049517119, 8.941666527059418, 9.348017938032347, 10.00994953494050, 10.74655470150871, 11.03979080178682, 11.43727442082865, 11.99437313111815, 12.65441188437685, 13.02568053235449, 13.41308044222175, 14.12348984511172, 14.69525566462333
