| L(s) = 1 | + 5·5-s − 11.6·7-s + 47.4·11-s + 84.2·13-s − 26.9·17-s + 152.·19-s + 177.·23-s + 25·25-s + 60.0·29-s + 92.1·31-s − 58.1·35-s − 221.·37-s + 115.·41-s − 383.·43-s − 317.·47-s − 207.·49-s − 257.·53-s + 237.·55-s + 642.·59-s − 662.·61-s + 421.·65-s − 597.·67-s + 500.·71-s + 989.·73-s − 552.·77-s + 517.·79-s + 605.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.627·7-s + 1.30·11-s + 1.79·13-s − 0.384·17-s + 1.83·19-s + 1.60·23-s + 0.200·25-s + 0.384·29-s + 0.533·31-s − 0.280·35-s − 0.985·37-s + 0.438·41-s − 1.36·43-s − 0.984·47-s − 0.605·49-s − 0.667·53-s + 0.582·55-s + 1.41·59-s − 1.38·61-s + 0.804·65-s − 1.08·67-s + 0.836·71-s + 1.58·73-s − 0.817·77-s + 0.737·79-s + 0.801·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.185628981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.185628981\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 + 11.6T + 343T^{2} \) |
| 11 | \( 1 - 47.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 26.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 152.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 60.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 92.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 221.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 115.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 383.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 317.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 257.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 642.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 662.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 597.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 500.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 989.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 517.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 605.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 742.T + 9.12e5T^{2} \) |
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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
−8.917972637610815892229257778084, −8.068675368471570727937619161780, −6.77847611362638145934252587530, −6.59345867772788218630422796037, −5.64491202948931594964854409044, −4.75296448736718724784634962258, −3.53469868496814765343216133620, −3.15383637333666551214407826293, −1.55807849613900719509430913990, −0.906706082680708087309809639864,
0.906706082680708087309809639864, 1.55807849613900719509430913990, 3.15383637333666551214407826293, 3.53469868496814765343216133620, 4.75296448736718724784634962258, 5.64491202948931594964854409044, 6.59345867772788218630422796037, 6.77847611362638145934252587530, 8.068675368471570727937619161780, 8.917972637610815892229257778084
