| L(s) = 1 | + 0.414·3-s − 1.82·5-s + 2.41·7-s − 2.82·9-s − 4.82·11-s − 13-s − 0.757·15-s + 5·17-s + 4.82·19-s + 0.999·21-s + 4·23-s − 1.65·25-s − 2.41·27-s + 3.65·29-s + 6·31-s − 1.99·33-s − 4.41·35-s + 8.65·37-s − 0.414·39-s + 8·41-s − 0.757·43-s + 5.17·45-s + 6.07·47-s − 1.17·49-s + 2.07·51-s + 8·53-s + 8.82·55-s + ⋯ |
| L(s) = 1 | + 0.239·3-s − 0.817·5-s + 0.912·7-s − 0.942·9-s − 1.45·11-s − 0.277·13-s − 0.195·15-s + 1.21·17-s + 1.10·19-s + 0.218·21-s + 0.834·23-s − 0.331·25-s − 0.464·27-s + 0.679·29-s + 1.07·31-s − 0.348·33-s − 0.746·35-s + 1.42·37-s − 0.0663·39-s + 1.24·41-s − 0.115·43-s + 0.770·45-s + 0.885·47-s − 0.167·49-s + 0.290·51-s + 1.09·53-s + 1.19·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.520604338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.520604338\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 5 | \( 1 + 1.82T + 5T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 8.65T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 0.757T + 43T^{2} \) |
| 47 | \( 1 - 6.07T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 + 8.82T + 67T^{2} \) |
| 71 | \( 1 + 9.72T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 7.31T + 79T^{2} \) |
| 83 | \( 1 - 6.34T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−9.286568758998919421058332970313, −8.286062598580559993329003632353, −7.81941045271336530465645938086, −7.42768674834865615961234272452, −5.91382593954788403661460861408, −5.23310946034240911318785057528, −4.45901218559489338733051481164, −3.19184527738422306677049168046, −2.56303139990172581193738959432, −0.848906116969302879647822827113,
0.848906116969302879647822827113, 2.56303139990172581193738959432, 3.19184527738422306677049168046, 4.45901218559489338733051481164, 5.23310946034240911318785057528, 5.91382593954788403661460861408, 7.42768674834865615961234272452, 7.81941045271336530465645938086, 8.286062598580559993329003632353, 9.286568758998919421058332970313
