| L(s) = 1 | + 0.539·2-s − 1.70·4-s + 5-s − 4.80·7-s − 2·8-s + 0.539·10-s + 3.41·11-s − 2.58·14-s + 2.34·16-s − 2.87·17-s + 4.97·19-s − 1.70·20-s + 1.84·22-s − 8.49·23-s + 25-s + 8.20·28-s + 3.51·29-s + 3.04·31-s + 5.26·32-s − 1.55·34-s − 4.80·35-s + 2.68·37-s + 2.68·38-s − 2·40-s + 3.75·41-s + 1.58·43-s − 5.84·44-s + ⋯ |
| L(s) = 1 | + 0.381·2-s − 0.854·4-s + 0.447·5-s − 1.81·7-s − 0.707·8-s + 0.170·10-s + 1.03·11-s − 0.691·14-s + 0.585·16-s − 0.698·17-s + 1.14·19-s − 0.382·20-s + 0.392·22-s − 1.77·23-s + 0.200·25-s + 1.55·28-s + 0.651·29-s + 0.547·31-s + 0.930·32-s − 0.266·34-s − 0.811·35-s + 0.440·37-s + 0.434·38-s − 0.316·40-s + 0.585·41-s + 0.242·43-s − 0.880·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.539T + 2T^{2} \) |
| 7 | \( 1 + 4.80T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 - 4.97T + 19T^{2} \) |
| 23 | \( 1 + 8.49T + 23T^{2} \) |
| 29 | \( 1 - 3.51T + 29T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 - 2.68T + 37T^{2} \) |
| 41 | \( 1 - 3.75T + 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 - 0.539T + 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 + 8.40T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 2.09T + 71T^{2} \) |
| 73 | \( 1 + 5.53T + 73T^{2} \) |
| 79 | \( 1 - 2.21T + 79T^{2} \) |
| 83 | \( 1 + 4.34T + 83T^{2} \) |
| 89 | \( 1 - 7.01T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.39623574638698020744838097870, −6.59818401917217992027021504501, −6.04746140376706071959037252585, −5.66366369023399146691569608703, −4.48506470932062598583507657351, −3.98128279762463713630898775522, −3.24345239431541316747935443713, −2.52495196853344125333118663198, −1.10043012316646547029900879188, 0,
1.10043012316646547029900879188, 2.52495196853344125333118663198, 3.24345239431541316747935443713, 3.98128279762463713630898775522, 4.48506470932062598583507657351, 5.66366369023399146691569608703, 6.04746140376706071959037252585, 6.59818401917217992027021504501, 7.39623574638698020744838097870
