| L(s) = 1 | − 1.64·3-s + 2.39·5-s − 7-s − 0.301·9-s − 11-s − 13-s − 3.94·15-s − 5.56·17-s + 1.34·19-s + 1.64·21-s + 2.38·23-s + 0.758·25-s + 5.42·27-s − 0.520·29-s + 8.43·31-s + 1.64·33-s − 2.39·35-s − 5.34·37-s + 1.64·39-s + 1.51·41-s + 8.44·43-s − 0.722·45-s − 0.408·47-s + 49-s + 9.13·51-s + 14.0·53-s − 2.39·55-s + ⋯ |
| L(s) = 1 | − 0.948·3-s + 1.07·5-s − 0.377·7-s − 0.100·9-s − 0.301·11-s − 0.277·13-s − 1.01·15-s − 1.34·17-s + 0.308·19-s + 0.358·21-s + 0.497·23-s + 0.151·25-s + 1.04·27-s − 0.0965·29-s + 1.51·31-s + 0.285·33-s − 0.405·35-s − 0.879·37-s + 0.263·39-s + 0.236·41-s + 1.28·43-s − 0.107·45-s − 0.0595·47-s + 0.142·49-s + 1.27·51-s + 1.93·53-s − 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.64T + 3T^{2} \) |
| 5 | \( 1 - 2.39T + 5T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 - 1.34T + 19T^{2} \) |
| 23 | \( 1 - 2.38T + 23T^{2} \) |
| 29 | \( 1 + 0.520T + 29T^{2} \) |
| 31 | \( 1 - 8.43T + 31T^{2} \) |
| 37 | \( 1 + 5.34T + 37T^{2} \) |
| 41 | \( 1 - 1.51T + 41T^{2} \) |
| 43 | \( 1 - 8.44T + 43T^{2} \) |
| 47 | \( 1 + 0.408T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 5.50T + 61T^{2} \) |
| 67 | \( 1 + 5.69T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 + 5.85T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 6.66T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.18479309924300091703951980756, −6.66819530048293680262416656066, −6.03162207653685069315989974348, −5.51970800897385692183558890490, −4.88214470244815746123843259396, −4.09874224102434783078593517560, −2.83621684703879289783438041372, −2.32571504312232413714910215847, −1.12970028524986526897660205189, 0,
1.12970028524986526897660205189, 2.32571504312232413714910215847, 2.83621684703879289783438041372, 4.09874224102434783078593517560, 4.88214470244815746123843259396, 5.51970800897385692183558890490, 6.03162207653685069315989974348, 6.66819530048293680262416656066, 7.18479309924300091703951980756
