| L(s) = 1 | − 3-s − 2·4-s − 2·5-s − 3·7-s + 9-s + 2·12-s + 2·15-s + 4·16-s − 2·17-s + 4·19-s + 4·20-s + 3·21-s − 4·23-s − 25-s − 27-s + 6·28-s − 5·31-s + 6·35-s − 2·36-s + 6·37-s − 8·41-s − 5·43-s − 2·45-s − 4·48-s + 2·49-s + 2·51-s + 14·53-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.894·5-s − 1.13·7-s + 1/3·9-s + 0.577·12-s + 0.516·15-s + 16-s − 0.485·17-s + 0.917·19-s + 0.894·20-s + 0.654·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.13·28-s − 0.898·31-s + 1.01·35-s − 1/3·36-s + 0.986·37-s − 1.24·41-s − 0.762·43-s − 0.298·45-s − 0.577·48-s + 2/7·49-s + 0.280·51-s + 1.92·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2265678536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2265678536\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.24362680169718, −13.58948359837593, −13.19299019177310, −12.93061993453998, −12.14772306988281, −11.82247642551027, −11.48357381093126, −10.54705676710846, −10.24550207928221, −9.619037183823976, −9.301261511655925, −8.598106122865377, −8.119370098996984, −7.483004745519456, −7.019256601213776, −6.373467692106210, −5.717502364232774, −5.335731554678028, −4.538730858190709, −4.072880542018339, −3.544458606378133, −3.072371649176004, −2.014663737589458, −0.9937471637151446, −0.2085895460199965,
0.2085895460199965, 0.9937471637151446, 2.014663737589458, 3.072371649176004, 3.544458606378133, 4.072880542018339, 4.538730858190709, 5.335731554678028, 5.717502364232774, 6.373467692106210, 7.019256601213776, 7.483004745519456, 8.119370098996984, 8.598106122865377, 9.301261511655925, 9.619037183823976, 10.24550207928221, 10.54705676710846, 11.48357381093126, 11.82247642551027, 12.14772306988281, 12.93061993453998, 13.19299019177310, 13.58948359837593, 14.24362680169718
