| L(s) = 1 | + 5-s − 2.73·7-s + 4.19·11-s − 4.73·13-s + 3.19·17-s + 2.26·19-s − 0.464·23-s + 25-s − 9.66·29-s − 9.19·31-s − 2.73·35-s − 7.46·37-s − 3.26·41-s + 3.26·43-s + 7.46·47-s + 0.464·49-s − 2.80·53-s + 4.19·55-s + 4.73·59-s + 11.9·61-s − 4.73·65-s + 0.928·67-s + 5.66·71-s + 0.196·73-s − 11.4·77-s + 14.1·79-s − 14.8·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.03·7-s + 1.26·11-s − 1.31·13-s + 0.775·17-s + 0.520·19-s − 0.0967·23-s + 0.200·25-s − 1.79·29-s − 1.65·31-s − 0.461·35-s − 1.22·37-s − 0.510·41-s + 0.498·43-s + 1.08·47-s + 0.0663·49-s − 0.385·53-s + 0.565·55-s + 0.616·59-s + 1.52·61-s − 0.586·65-s + 0.113·67-s + 0.671·71-s + 0.0229·73-s − 1.30·77-s + 1.58·79-s − 1.63·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 - 4.19T + 11T^{2} \) |
| 13 | \( 1 + 4.73T + 13T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 19 | \( 1 - 2.26T + 19T^{2} \) |
| 23 | \( 1 + 0.464T + 23T^{2} \) |
| 29 | \( 1 + 9.66T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 + 7.46T + 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 - 3.26T + 43T^{2} \) |
| 47 | \( 1 - 7.46T + 47T^{2} \) |
| 53 | \( 1 + 2.80T + 53T^{2} \) |
| 59 | \( 1 - 4.73T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 0.928T + 67T^{2} \) |
| 71 | \( 1 - 5.66T + 71T^{2} \) |
| 73 | \( 1 - 0.196T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 1.46T + 97T^{2} \) |
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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
−7.921241425092311591720054153332, −7.01466699710500967991293185219, −6.81605679435334625598929201639, −5.60293322024917231365789628723, −5.36572754146350749244203889348, −3.98452354267102359517795246236, −3.48885521211283544508025469288, −2.44077145671352364575613625238, −1.45193648404109700889204593146, 0,
1.45193648404109700889204593146, 2.44077145671352364575613625238, 3.48885521211283544508025469288, 3.98452354267102359517795246236, 5.36572754146350749244203889348, 5.60293322024917231365789628723, 6.81605679435334625598929201639, 7.01466699710500967991293185219, 7.921241425092311591720054153332
