| L(s) = 1 | − 2-s + 4-s − 2.80·5-s − 8-s + 2.80·10-s + 11-s + 16-s + 7.96·17-s + 6.48·19-s − 2.80·20-s − 22-s + 5.28·23-s + 2.87·25-s − 5.12·29-s + 8.96·31-s − 32-s − 7.96·34-s + 10.4·37-s − 6.48·38-s + 2.80·40-s − 5.09·41-s + 11.4·43-s + 44-s − 5.28·46-s + 0.322·47-s − 2.87·50-s − 8·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.25·5-s − 0.353·8-s + 0.887·10-s + 0.301·11-s + 0.250·16-s + 1.93·17-s + 1.48·19-s − 0.627·20-s − 0.213·22-s + 1.10·23-s + 0.574·25-s − 0.952·29-s + 1.61·31-s − 0.176·32-s − 1.36·34-s + 1.72·37-s − 1.05·38-s + 0.443·40-s − 0.795·41-s + 1.74·43-s + 0.150·44-s − 0.779·46-s + 0.0471·47-s − 0.406·50-s − 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.401588979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.401588979\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2.80T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 7.96T + 17T^{2} \) |
| 19 | \( 1 - 6.48T + 19T^{2} \) |
| 23 | \( 1 - 5.28T + 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 - 8.96T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 5.09T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 0.322T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + 0.871T + 59T^{2} \) |
| 61 | \( 1 + 2.15T + 61T^{2} \) |
| 67 | \( 1 + 7.09T + 67T^{2} \) |
| 71 | \( 1 - 7.44T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 - 6.80T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 1.61T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
| 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
−7.74058182651774404112913926360, −7.38160164959207173344265217663, −6.46838428320315215235015269043, −5.69192670022197182700722860288, −4.94525939477218560836992475006, −4.04890950919402590687095102181, −3.29043204129025094557510511936, −2.79646670768548651461284563831, −1.28994298532976653712006844945, −0.73283151413975283700636296459,
0.73283151413975283700636296459, 1.28994298532976653712006844945, 2.79646670768548651461284563831, 3.29043204129025094557510511936, 4.04890950919402590687095102181, 4.94525939477218560836992475006, 5.69192670022197182700722860288, 6.46838428320315215235015269043, 7.38160164959207173344265217663, 7.74058182651774404112913926360
