| L(s) = 1 | + 2·5-s − 3·9-s − 4·11-s + 2·13-s + 2·17-s − 4·19-s − 4·23-s − 25-s − 6·29-s − 8·31-s − 10·37-s + 2·41-s − 4·43-s − 6·45-s − 7·49-s + 10·53-s − 8·55-s − 4·59-s − 2·61-s + 4·65-s + 12·71-s + 73-s − 4·79-s + 9·81-s − 12·83-s + 4·85-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 1.64·37-s + 0.312·41-s − 0.609·43-s − 0.894·45-s − 49-s + 1.37·53-s − 1.07·55-s − 0.520·59-s − 0.256·61-s + 0.496·65-s + 1.42·71-s + 0.117·73-s − 0.450·79-s + 81-s − 1.31·83-s + 0.433·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 73 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.67749789687173, −13.34594968863159, −12.85178147241504, −12.45621143702301, −11.84172476565304, −11.28052895329267, −10.84899975018173, −10.45483648947341, −9.984428943043654, −9.476884862842863, −8.872423881977207, −8.571574443758790, −7.884347061272482, −7.663725604243252, −6.779572462004252, −6.395102195417550, −5.622661126235009, −5.528967254693890, −5.145557531433415, −4.156072738455356, −3.661625936203580, −3.082820940512198, −2.372592672941068, −1.980675178147221, −1.379289117812820, 0, 0,
1.379289117812820, 1.980675178147221, 2.372592672941068, 3.082820940512198, 3.661625936203580, 4.156072738455356, 5.145557531433415, 5.528967254693890, 5.622661126235009, 6.395102195417550, 6.779572462004252, 7.663725604243252, 7.884347061272482, 8.571574443758790, 8.872423881977207, 9.476884862842863, 9.984428943043654, 10.45483648947341, 10.84899975018173, 11.28052895329267, 11.84172476565304, 12.45621143702301, 12.85178147241504, 13.34594968863159, 13.67749789687173
