| L(s) = 1 | + 2.57·3-s + 0.350·5-s − 7-s + 3.60·9-s + 1.25·11-s + 6.92·13-s + 0.900·15-s − 2.25·17-s + 5.55·19-s − 2.57·21-s − 6.22·23-s − 4.87·25-s + 1.56·27-s − 5.45·29-s + 10.2·31-s + 3.21·33-s − 0.350·35-s + 2.17·37-s + 17.8·39-s − 41-s + 1.54·43-s + 1.26·45-s + 5.11·47-s + 49-s − 5.80·51-s + 10.0·53-s + 0.438·55-s + ⋯ |
| L(s) = 1 | + 1.48·3-s + 0.156·5-s − 0.377·7-s + 1.20·9-s + 0.377·11-s + 1.92·13-s + 0.232·15-s − 0.547·17-s + 1.27·19-s − 0.561·21-s − 1.29·23-s − 0.975·25-s + 0.301·27-s − 1.01·29-s + 1.83·31-s + 0.559·33-s − 0.0592·35-s + 0.357·37-s + 2.85·39-s − 0.156·41-s + 0.235·43-s + 0.188·45-s + 0.746·47-s + 0.142·49-s − 0.813·51-s + 1.37·53-s + 0.0590·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.897269417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.897269417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 2.57T + 3T^{2} \) |
| 5 | \( 1 - 0.350T + 5T^{2} \) |
| 11 | \( 1 - 1.25T + 11T^{2} \) |
| 13 | \( 1 - 6.92T + 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 - 5.55T + 19T^{2} \) |
| 23 | \( 1 + 6.22T + 23T^{2} \) |
| 29 | \( 1 + 5.45T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 2.17T + 37T^{2} \) |
| 43 | \( 1 - 1.54T + 43T^{2} \) |
| 47 | \( 1 - 5.11T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 1.51T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 16.5T + 71T^{2} \) |
| 73 | \( 1 - 5.30T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 5.46T + 83T^{2} \) |
| 89 | \( 1 - 6.80T + 89T^{2} \) |
| 97 | \( 1 + 3.91T + 97T^{2} \) |
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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
−9.534064917741377265881371382596, −9.014528997056351956409429818137, −8.216228137637033087570215772381, −7.62932755983311858165073336130, −6.46210078396355131626955973236, −5.74859079373776658707199457200, −4.11349890133287350400221543111, −3.60567809831663301030235353599, −2.57502558321885427191247256243, −1.40949214868331680838551481947,
1.40949214868331680838551481947, 2.57502558321885427191247256243, 3.60567809831663301030235353599, 4.11349890133287350400221543111, 5.74859079373776658707199457200, 6.46210078396355131626955973236, 7.62932755983311858165073336130, 8.216228137637033087570215772381, 9.014528997056351956409429818137, 9.534064917741377265881371382596
