| L(s) = 1 | − 1.47·2-s + 0.182·4-s − 5-s + 2.29·7-s + 2.68·8-s + 1.47·10-s + 2.14·13-s − 3.39·14-s − 4.33·16-s + 0.544·17-s + 2.18·19-s − 0.182·20-s − 2.03·23-s + 25-s − 3.16·26-s + 0.418·28-s − 9.94·29-s + 6.77·31-s + 1.02·32-s − 0.804·34-s − 2.29·35-s − 8.81·37-s − 3.22·38-s − 2.68·40-s − 1.82·41-s + 0.620·43-s + 3.01·46-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.0911·4-s − 0.447·5-s + 0.867·7-s + 0.949·8-s + 0.467·10-s + 0.593·13-s − 0.906·14-s − 1.08·16-s + 0.132·17-s + 0.500·19-s − 0.0407·20-s − 0.425·23-s + 0.200·25-s − 0.620·26-s + 0.0790·28-s − 1.84·29-s + 1.21·31-s + 0.181·32-s − 0.137·34-s − 0.387·35-s − 1.44·37-s − 0.522·38-s − 0.424·40-s − 0.284·41-s + 0.0946·43-s + 0.444·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.47T + 2T^{2} \) |
| 7 | \( 1 - 2.29T + 7T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 - 0.544T + 17T^{2} \) |
| 19 | \( 1 - 2.18T + 19T^{2} \) |
| 23 | \( 1 + 2.03T + 23T^{2} \) |
| 29 | \( 1 + 9.94T + 29T^{2} \) |
| 31 | \( 1 - 6.77T + 31T^{2} \) |
| 37 | \( 1 + 8.81T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 - 0.620T + 43T^{2} \) |
| 47 | \( 1 - 0.378T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 8.07T + 59T^{2} \) |
| 61 | \( 1 + 8.72T + 61T^{2} \) |
| 67 | \( 1 + 9.75T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 7.85T + 73T^{2} \) |
| 79 | \( 1 - 9.22T + 79T^{2} \) |
| 83 | \( 1 + 9.45T + 83T^{2} \) |
| 89 | \( 1 + 0.583T + 89T^{2} \) |
| 97 | \( 1 + 5.37T + 97T^{2} \) |
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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
−7.959704767506744893810750127392, −7.43953111743902847367355785813, −6.63810909966374037572873007696, −5.59466555401448692156068221026, −4.86543543139863851285295400014, −4.12593547986538733984048277707, −3.29240755448038010548326197758, −1.93159533831963081943074369533, −1.23564031931181568632074694770, 0,
1.23564031931181568632074694770, 1.93159533831963081943074369533, 3.29240755448038010548326197758, 4.12593547986538733984048277707, 4.86543543139863851285295400014, 5.59466555401448692156068221026, 6.63810909966374037572873007696, 7.43953111743902847367355785813, 7.959704767506744893810750127392
