| L(s) = 1 | + 3.40·5-s − 7-s + 1.23·11-s − 6.47·13-s − 17-s − 2.21·19-s − 1.39·23-s + 6.60·25-s + 0.633·29-s + 0.965·31-s − 3.40·35-s − 8.31·37-s + 5.60·41-s + 11.0·43-s − 7.67·47-s + 49-s − 11.7·53-s + 4.21·55-s + 0.602·59-s − 9.84·61-s − 22.0·65-s + 5.30·67-s + 3.33·71-s + 14.0·73-s − 1.23·77-s − 0.323·79-s − 13.8·83-s + ⋯ |
| L(s) = 1 | + 1.52·5-s − 0.377·7-s + 0.372·11-s − 1.79·13-s − 0.242·17-s − 0.507·19-s − 0.291·23-s + 1.32·25-s + 0.117·29-s + 0.173·31-s − 0.575·35-s − 1.36·37-s + 0.875·41-s + 1.68·43-s − 1.11·47-s + 0.142·49-s − 1.60·53-s + 0.567·55-s + 0.0783·59-s − 1.26·61-s − 2.73·65-s + 0.648·67-s + 0.395·71-s + 1.64·73-s − 0.140·77-s − 0.0363·79-s − 1.52·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 3.40T + 5T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 + 1.39T + 23T^{2} \) |
| 29 | \( 1 - 0.633T + 29T^{2} \) |
| 31 | \( 1 - 0.965T + 31T^{2} \) |
| 37 | \( 1 + 8.31T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 7.67T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 0.602T + 59T^{2} \) |
| 61 | \( 1 + 9.84T + 61T^{2} \) |
| 67 | \( 1 - 5.30T + 67T^{2} \) |
| 71 | \( 1 - 3.33T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 0.323T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 5.90T + 97T^{2} \) |
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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
−7.34212744667581343318603622304, −6.55744469605996232872292578074, −6.20765642425511268051227327621, −5.30734151011947283428423597336, −4.84554660397811510125439355708, −3.90673796767910434427550706185, −2.76225651945794503048407407029, −2.29347647688561400306008408366, −1.43706097837936883742681352679, 0,
1.43706097837936883742681352679, 2.29347647688561400306008408366, 2.76225651945794503048407407029, 3.90673796767910434427550706185, 4.84554660397811510125439355708, 5.30734151011947283428423597336, 6.20765642425511268051227327621, 6.55744469605996232872292578074, 7.34212744667581343318603622304
