| L(s) = 1 | − 0.161·3-s + 5-s − 0.359·7-s − 2.97·9-s + 4.96·11-s + 5.13·13-s − 0.161·15-s + 4.76·17-s + 4.84·19-s + 0.0578·21-s + 7.93·23-s + 25-s + 0.962·27-s − 3.70·29-s − 0.925·31-s − 0.800·33-s − 0.359·35-s − 4.26·37-s − 0.827·39-s + 5.39·41-s + 11.4·43-s − 2.97·45-s − 9.45·47-s − 6.87·49-s − 0.767·51-s + 1.52·53-s + 4.96·55-s + ⋯ |
| L(s) = 1 | − 0.0930·3-s + 0.447·5-s − 0.135·7-s − 0.991·9-s + 1.49·11-s + 1.42·13-s − 0.0416·15-s + 1.15·17-s + 1.11·19-s + 0.0126·21-s + 1.65·23-s + 0.200·25-s + 0.185·27-s − 0.688·29-s − 0.166·31-s − 0.139·33-s − 0.0607·35-s − 0.701·37-s − 0.132·39-s + 0.842·41-s + 1.73·43-s − 0.443·45-s − 1.37·47-s − 0.981·49-s − 0.107·51-s + 0.209·53-s + 0.669·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.793495074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.793495074\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 0.161T + 3T^{2} \) |
| 7 | \( 1 + 0.359T + 7T^{2} \) |
| 11 | \( 1 - 4.96T + 11T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 17 | \( 1 - 4.76T + 17T^{2} \) |
| 19 | \( 1 - 4.84T + 19T^{2} \) |
| 23 | \( 1 - 7.93T + 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 0.925T + 31T^{2} \) |
| 37 | \( 1 + 4.26T + 37T^{2} \) |
| 41 | \( 1 - 5.39T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 9.45T + 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 1.66T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 0.552T + 71T^{2} \) |
| 73 | \( 1 + 5.16T + 73T^{2} \) |
| 79 | \( 1 - 8.46T + 79T^{2} \) |
| 83 | \( 1 + 9.93T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 0.0798T + 97T^{2} \) |
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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
−7.85781069368031779930526886142, −7.00940389070652838193689470760, −6.42123317826336442172599031578, −5.65525962952123388537121803455, −5.36742565291083882902988275213, −4.15485247608873433774982795867, −3.40630475480747475605610697575, −2.90163454435173794999219862972, −1.49867927271116499765142958036, −0.946543817481939243359266128711,
0.946543817481939243359266128711, 1.49867927271116499765142958036, 2.90163454435173794999219862972, 3.40630475480747475605610697575, 4.15485247608873433774982795867, 5.36742565291083882902988275213, 5.65525962952123388537121803455, 6.42123317826336442172599031578, 7.00940389070652838193689470760, 7.85781069368031779930526886142
