| L(s) = 1 | + 0.409·2-s + 1.91·3-s − 1.83·4-s + 0.785·6-s + 3.66·7-s − 1.56·8-s + 0.682·9-s + 5.60·11-s − 3.51·12-s − 4.56·13-s + 1.49·14-s + 3.02·16-s − 6.06·17-s + 0.279·18-s − 3.67·19-s + 7.02·21-s + 2.29·22-s − 7.36·23-s − 3.01·24-s − 1.86·26-s − 4.44·27-s − 6.70·28-s − 1.37·29-s + 2.62·31-s + 4.37·32-s + 10.7·33-s − 2.48·34-s + ⋯ |
| L(s) = 1 | + 0.289·2-s + 1.10·3-s − 0.916·4-s + 0.320·6-s + 1.38·7-s − 0.554·8-s + 0.227·9-s + 1.68·11-s − 1.01·12-s − 1.26·13-s + 0.400·14-s + 0.755·16-s − 1.47·17-s + 0.0658·18-s − 0.843·19-s + 1.53·21-s + 0.489·22-s − 1.53·23-s − 0.614·24-s − 0.366·26-s − 0.855·27-s − 1.26·28-s − 0.255·29-s + 0.471·31-s + 0.773·32-s + 1.87·33-s − 0.425·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 0.409T + 2T^{2} \) |
| 3 | \( 1 - 1.91T + 3T^{2} \) |
| 7 | \( 1 - 3.66T + 7T^{2} \) |
| 11 | \( 1 - 5.60T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 17 | \( 1 + 6.06T + 17T^{2} \) |
| 19 | \( 1 + 3.67T + 19T^{2} \) |
| 23 | \( 1 + 7.36T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 + 3.20T + 37T^{2} \) |
| 41 | \( 1 + 9.05T + 41T^{2} \) |
| 43 | \( 1 + 8.36T + 43T^{2} \) |
| 47 | \( 1 + 8.91T + 47T^{2} \) |
| 53 | \( 1 + 8.02T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 6.04T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 4.63T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 + 0.0114T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.158904609886184588309263108552, −7.05080267801600515264189356269, −6.39298039115697081435159307164, −5.31032170477327646199343757351, −4.58992459126944912986080435922, −4.13235964156504608435701110693, −3.42054870877257258848024685553, −2.19442980133392477871409068011, −1.70401399045635940458032601087, 0,
1.70401399045635940458032601087, 2.19442980133392477871409068011, 3.42054870877257258848024685553, 4.13235964156504608435701110693, 4.58992459126944912986080435922, 5.31032170477327646199343757351, 6.39298039115697081435159307164, 7.05080267801600515264189356269, 8.158904609886184588309263108552
