| L(s) = 1 | + 1.08·2-s + 2.92·3-s − 0.816·4-s + 5-s + 3.18·6-s + 7-s − 3.06·8-s + 5.55·9-s + 1.08·10-s − 4.47·11-s − 2.38·12-s − 4.40·13-s + 1.08·14-s + 2.92·15-s − 1.70·16-s + 4.19·17-s + 6.04·18-s − 0.942·19-s − 0.816·20-s + 2.92·21-s − 4.86·22-s − 9.16·23-s − 8.96·24-s + 25-s − 4.79·26-s + 7.48·27-s − 0.816·28-s + ⋯ |
| L(s) = 1 | + 0.769·2-s + 1.68·3-s − 0.408·4-s + 0.447·5-s + 1.29·6-s + 0.377·7-s − 1.08·8-s + 1.85·9-s + 0.344·10-s − 1.34·11-s − 0.689·12-s − 1.22·13-s + 0.290·14-s + 0.755·15-s − 0.425·16-s + 1.01·17-s + 1.42·18-s − 0.216·19-s − 0.182·20-s + 0.638·21-s − 1.03·22-s − 1.91·23-s − 1.82·24-s + 0.200·25-s − 0.940·26-s + 1.44·27-s − 0.154·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 1.08T + 2T^{2} \) |
| 3 | \( 1 - 2.92T + 3T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 + 4.40T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 + 0.942T + 19T^{2} \) |
| 23 | \( 1 + 9.16T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 3.26T + 43T^{2} \) |
| 47 | \( 1 - 4.33T + 47T^{2} \) |
| 53 | \( 1 + 3.15T + 53T^{2} \) |
| 59 | \( 1 - 0.293T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 1.77T + 73T^{2} \) |
| 79 | \( 1 + 5.78T + 79T^{2} \) |
| 83 | \( 1 + 8.76T + 83T^{2} \) |
| 89 | \( 1 + 5.53T + 89T^{2} \) |
| 97 | \( 1 - 2.19T + 97T^{2} \) |
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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
−7.71145611321416952699240902897, −7.02333670581814488044511087388, −5.70481662253027962445601049863, −5.38671564317085132406884707588, −4.53578715573470202069539444850, −3.78228688813299671349771062247, −3.19305772224155936493831614452, −2.34694390689933300197052622040, −1.88159162554334209849669775800, 0,
1.88159162554334209849669775800, 2.34694390689933300197052622040, 3.19305772224155936493831614452, 3.78228688813299671349771062247, 4.53578715573470202069539444850, 5.38671564317085132406884707588, 5.70481662253027962445601049863, 7.02333670581814488044511087388, 7.71145611321416952699240902897
