| L(s) = 1 | − 1.33·2-s − 0.216·4-s − 1.70·5-s − 3.82·7-s + 2.96·8-s + 2.27·10-s − 11-s − 1.41·13-s + 5.10·14-s − 3.51·16-s + 2.70·17-s − 2.25·19-s + 0.369·20-s + 1.33·22-s − 2.60·23-s − 2.09·25-s + 1.88·26-s + 0.829·28-s + 5.40·29-s + 1.94·31-s − 1.22·32-s − 3.61·34-s + 6.51·35-s + 2.07·37-s + 3.01·38-s − 5.04·40-s + 1.68·41-s + ⋯ |
| L(s) = 1 | − 0.944·2-s − 0.108·4-s − 0.762·5-s − 1.44·7-s + 1.04·8-s + 0.719·10-s − 0.301·11-s − 0.392·13-s + 1.36·14-s − 0.879·16-s + 0.656·17-s − 0.517·19-s + 0.0826·20-s + 0.284·22-s − 0.542·23-s − 0.418·25-s + 0.370·26-s + 0.156·28-s + 1.00·29-s + 0.348·31-s − 0.215·32-s − 0.619·34-s + 1.10·35-s + 0.341·37-s + 0.488·38-s − 0.798·40-s + 0.263·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.33T + 2T^{2} \) |
| 5 | \( 1 + 1.70T + 5T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 2.70T + 17T^{2} \) |
| 19 | \( 1 + 2.25T + 19T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 - 5.40T + 29T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 37 | \( 1 - 2.07T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 - 4.50T + 43T^{2} \) |
| 47 | \( 1 - 2.55T + 47T^{2} \) |
| 53 | \( 1 - 5.43T + 53T^{2} \) |
| 59 | \( 1 + 4.50T + 59T^{2} \) |
| 67 | \( 1 - 1.34T + 67T^{2} \) |
| 71 | \( 1 - 2.13T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 3.83T + 79T^{2} \) |
| 83 | \( 1 - 7.05T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 7.63T + 97T^{2} \) |
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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
−7.77406198066913449957917234692, −7.29422834751081886195222292496, −6.46263615422840597914502812510, −5.73953804344956224787737945513, −4.66631924246323945135492629676, −4.00933210629553042197863804191, −3.21722293820790248802248871950, −2.27205590779438409591258631760, −0.843283703876702211763189323884, 0,
0.843283703876702211763189323884, 2.27205590779438409591258631760, 3.21722293820790248802248871950, 4.00933210629553042197863804191, 4.66631924246323945135492629676, 5.73953804344956224787737945513, 6.46263615422840597914502812510, 7.29422834751081886195222292496, 7.77406198066913449957917234692
