| L(s) = 1 | − 2-s + 4-s − 2.73·5-s + 7-s − 8-s + 2.73·10-s + 3.46·11-s − 1.46·13-s − 14-s + 16-s + 0.732·17-s − 2·19-s − 2.73·20-s − 3.46·22-s − 23-s + 2.46·25-s + 1.46·26-s + 28-s − 8·29-s + 0.196·31-s − 32-s − 0.732·34-s − 2.73·35-s + 0.535·37-s + 2·38-s + 2.73·40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.22·5-s + 0.377·7-s − 0.353·8-s + 0.863·10-s + 1.04·11-s − 0.406·13-s − 0.267·14-s + 0.250·16-s + 0.177·17-s − 0.458·19-s − 0.610·20-s − 0.738·22-s − 0.208·23-s + 0.492·25-s + 0.287·26-s + 0.188·28-s − 1.48·29-s + 0.0352·31-s − 0.176·32-s − 0.125·34-s − 0.461·35-s + 0.0881·37-s + 0.324·38-s + 0.431·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 - 0.732T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 0.196T + 31T^{2} \) |
| 37 | \( 1 - 0.535T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 + 4.53T + 67T^{2} \) |
| 71 | \( 1 + 8.39T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 - 0.928T + 79T^{2} \) |
| 83 | \( 1 - 6.39T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 8.73T + 97T^{2} \) |
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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
−8.418385845216893773065805396349, −7.61366718325074957265050904203, −7.21122372819985152531563685800, −6.30020894599441998167433158157, −5.34096492615294192951695102539, −4.17762520132954813782859672028, −3.73662718221566422543883226726, −2.47046372628679781854104138469, −1.30279597213553672093675431844, 0,
1.30279597213553672093675431844, 2.47046372628679781854104138469, 3.73662718221566422543883226726, 4.17762520132954813782859672028, 5.34096492615294192951695102539, 6.30020894599441998167433158157, 7.21122372819985152531563685800, 7.61366718325074957265050904203, 8.418385845216893773065805396349
