| L(s) = 1 | − 2.70·2-s + 5.31·4-s + 5-s + 3.91·7-s − 8.96·8-s − 2.70·10-s − 0.290·11-s − 2.36·13-s − 10.5·14-s + 13.6·16-s − 4.13·17-s − 7.64·19-s + 5.31·20-s + 0.786·22-s + 6.01·23-s + 25-s + 6.38·26-s + 20.8·28-s − 8.87·29-s + 7.33·31-s − 18.8·32-s + 11.1·34-s + 3.91·35-s + 0.927·37-s + 20.6·38-s − 8.96·40-s + 10.6·41-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 2.65·4-s + 0.447·5-s + 1.47·7-s − 3.16·8-s − 0.855·10-s − 0.0876·11-s − 0.654·13-s − 2.82·14-s + 3.40·16-s − 1.00·17-s − 1.75·19-s + 1.18·20-s + 0.167·22-s + 1.25·23-s + 0.200·25-s + 1.25·26-s + 3.93·28-s − 1.64·29-s + 1.31·31-s − 3.33·32-s + 1.91·34-s + 0.661·35-s + 0.152·37-s + 3.35·38-s − 1.41·40-s + 1.65·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 7 | \( 1 - 3.91T + 7T^{2} \) |
| 11 | \( 1 + 0.290T + 11T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 17 | \( 1 + 4.13T + 17T^{2} \) |
| 19 | \( 1 + 7.64T + 19T^{2} \) |
| 23 | \( 1 - 6.01T + 23T^{2} \) |
| 29 | \( 1 + 8.87T + 29T^{2} \) |
| 31 | \( 1 - 7.33T + 31T^{2} \) |
| 37 | \( 1 - 0.927T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 9.40T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 4.74T + 53T^{2} \) |
| 59 | \( 1 - 8.49T + 59T^{2} \) |
| 61 | \( 1 + 2.98T + 61T^{2} \) |
| 67 | \( 1 + 7.70T + 67T^{2} \) |
| 71 | \( 1 + 4.65T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 3.43T + 83T^{2} \) |
| 97 | \( 1 + 6.74T + 97T^{2} \) |
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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
−8.179539052756361426552746304452, −7.67230981408227462928022358232, −6.83573102866436146046322652771, −6.28575452723751120671821332835, −5.21439807196008334622155126312, −4.36882025202460884464501268434, −2.75718203875530961435318758074, −2.05737703256643868471677590723, −1.39203256595251828815450149120, 0,
1.39203256595251828815450149120, 2.05737703256643868471677590723, 2.75718203875530961435318758074, 4.36882025202460884464501268434, 5.21439807196008334622155126312, 6.28575452723751120671821332835, 6.83573102866436146046322652771, 7.67230981408227462928022358232, 8.179539052756361426552746304452
