| L(s) = 1 | + 3.46·2-s + 3.99·4-s + 13.8·5-s + 22.5·7-s − 13.8·8-s + 47.9·10-s − 22.5·11-s + 77.9·14-s − 80·16-s + 27·17-s + 88.3·19-s + 55.4·20-s − 77.9·22-s + 57·23-s + 66.9·25-s + 90.0·28-s + 69·29-s + 72.7·31-s − 166.·32-s + 93.5·34-s + 311.·35-s + 39.8·37-s + 306·38-s − 192.·40-s + 393.·41-s + 85·43-s − 90.0·44-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s + 1.23·5-s + 1.21·7-s − 0.612·8-s + 1.51·10-s − 0.617·11-s + 1.48·14-s − 1.25·16-s + 0.385·17-s + 1.06·19-s + 0.619·20-s − 0.755·22-s + 0.516·23-s + 0.535·25-s + 0.607·28-s + 0.441·29-s + 0.421·31-s − 0.918·32-s + 0.471·34-s + 1.50·35-s + 0.177·37-s + 1.30·38-s − 0.758·40-s + 1.49·41-s + 0.301·43-s − 0.308·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.981542254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.981542254\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3.46T + 8T^{2} \) |
| 5 | \( 1 - 13.8T + 125T^{2} \) |
| 7 | \( 1 - 22.5T + 343T^{2} \) |
| 11 | \( 1 + 22.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 27T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 57T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69T + 2.43e4T^{2} \) |
| 31 | \( 1 - 72.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 39.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 85T + 7.95e4T^{2} \) |
| 47 | \( 1 - 342.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 426T + 1.48e5T^{2} \) |
| 59 | \( 1 - 19.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 17T + 2.26e5T^{2} \) |
| 67 | \( 1 - 164.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 583.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 426.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 306.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−9.218883539742072980589050618652, −8.240642238787495625090132914099, −7.37844690666233671496484335042, −6.27710749166735797984388683560, −5.51642712471554735263890513365, −5.10027356250848282585715401493, −4.26461854956984945155441710309, −3.01236119733689018606869116963, −2.22929072781959910677456522098, −1.05552906967875329890587494365,
1.05552906967875329890587494365, 2.22929072781959910677456522098, 3.01236119733689018606869116963, 4.26461854956984945155441710309, 5.10027356250848282585715401493, 5.51642712471554735263890513365, 6.27710749166735797984388683560, 7.37844690666233671496484335042, 8.240642238787495625090132914099, 9.218883539742072980589050618652
