| L(s) = 1 | − 3-s − 2·7-s + 9-s + 11-s − 4·13-s − 6·17-s + 4·19-s + 2·21-s − 6·23-s − 5·25-s − 27-s + 6·29-s − 8·31-s − 33-s − 10·37-s + 4·39-s + 6·41-s − 8·43-s + 6·47-s − 3·49-s + 6·51-s − 4·57-s + 8·61-s − 2·63-s + 4·67-s + 6·69-s − 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 1.45·17-s + 0.917·19-s + 0.436·21-s − 1.25·23-s − 25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.174·33-s − 1.64·37-s + 0.640·39-s + 0.937·41-s − 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.840·51-s − 0.529·57-s + 1.02·61-s − 0.251·63-s + 0.488·67-s + 0.722·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−10.28397817499553530530964506775, −9.702896608623430445687767205931, −8.767992898257258380407868928000, −7.47999694363717860882978383906, −6.73373984274544568686272664379, −5.80771575321423739175967870457, −4.75034201814524111107553764909, −3.62485072881439717663109857391, −2.12574870769826326935834056810, 0,
2.12574870769826326935834056810, 3.62485072881439717663109857391, 4.75034201814524111107553764909, 5.80771575321423739175967870457, 6.73373984274544568686272664379, 7.47999694363717860882978383906, 8.767992898257258380407868928000, 9.702896608623430445687767205931, 10.28397817499553530530964506775
