| L(s) = 1 | − 3.21·2-s − 3·3-s + 2.33·4-s + 2.50·5-s + 9.64·6-s + 5.55·7-s + 18.2·8-s + 9·9-s − 8.06·10-s − 11·11-s − 7.01·12-s + 53.0·13-s − 17.8·14-s − 7.52·15-s − 77.2·16-s − 92.3·17-s − 28.9·18-s + 4.84·19-s + 5.86·20-s − 16.6·21-s + 35.3·22-s + 178.·23-s − 54.6·24-s − 118.·25-s − 170.·26-s − 27·27-s + 12.9·28-s + ⋯ |
| L(s) = 1 | − 1.13·2-s − 0.577·3-s + 0.292·4-s + 0.224·5-s + 0.656·6-s + 0.299·7-s + 0.804·8-s + 0.333·9-s − 0.254·10-s − 0.301·11-s − 0.168·12-s + 1.13·13-s − 0.340·14-s − 0.129·15-s − 1.20·16-s − 1.31·17-s − 0.378·18-s + 0.0585·19-s + 0.0655·20-s − 0.173·21-s + 0.342·22-s + 1.61·23-s − 0.464·24-s − 0.949·25-s − 1.28·26-s − 0.192·27-s + 0.0876·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 3.21T + 8T^{2} \) |
| 5 | \( 1 - 2.50T + 125T^{2} \) |
| 7 | \( 1 - 5.55T + 343T^{2} \) |
| 13 | \( 1 - 53.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 92.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.84T + 6.85e3T^{2} \) |
| 23 | \( 1 - 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 282.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 37.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 388.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 415.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 513.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 208.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 482.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 580.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 44.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 287.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 280.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 915.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 712.T + 9.12e5T^{2} \) |
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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
−8.456576221987895299929345079977, −7.936689512481229516683143350561, −6.69527381570752018710418810208, −6.45461815243261156711496305856, −5.00530352865040239029109864275, −4.62889389948593961222144324799, −3.26359592182815510529465451477, −1.87918290367175150610172001768, −1.06368482855031685124487837648, 0,
1.06368482855031685124487837648, 1.87918290367175150610172001768, 3.26359592182815510529465451477, 4.62889389948593961222144324799, 5.00530352865040239029109864275, 6.45461815243261156711496305856, 6.69527381570752018710418810208, 7.936689512481229516683143350561, 8.456576221987895299929345079977
