| L(s) = 1 | − 3-s − 5-s + 7-s − 2·9-s + 11-s + 2·13-s + 15-s − 3·17-s + 19-s − 21-s − 6·23-s + 25-s + 5·27-s − 9·29-s − 5·31-s − 33-s − 35-s + 5·37-s − 2·39-s − 6·41-s − 8·43-s + 2·45-s − 6·47-s − 6·49-s + 3·51-s + 9·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.554·13-s + 0.258·15-s − 0.727·17-s + 0.229·19-s − 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s − 1.67·29-s − 0.898·31-s − 0.174·33-s − 0.169·35-s + 0.821·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s + 0.298·45-s − 0.875·47-s − 6/7·49-s + 0.420·51-s + 1.23·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.755174849293333018338487861010, −8.760882245212911918603642776160, −8.117642961319108642473978475084, −7.10601287556001172013851251569, −6.15105215702104870707666270277, −5.39760376933933467243261462675, −4.32911684200491106989841650193, −3.34463148493755507208703338313, −1.80534900183808451831796459613, 0,
1.80534900183808451831796459613, 3.34463148493755507208703338313, 4.32911684200491106989841650193, 5.39760376933933467243261462675, 6.15105215702104870707666270277, 7.10601287556001172013851251569, 8.117642961319108642473978475084, 8.760882245212911918603642776160, 9.755174849293333018338487861010
