| L(s) = 1 | + 1.31·2-s − 3-s − 0.267·4-s − 1.10·5-s − 1.31·6-s + 1.45·7-s − 2.98·8-s + 9-s − 1.45·10-s − 0.837·11-s + 0.267·12-s + 0.345·13-s + 1.91·14-s + 1.10·15-s − 3.39·16-s + 5.10·17-s + 1.31·18-s + 5.49·19-s + 0.296·20-s − 1.45·21-s − 1.10·22-s − 1.19·23-s + 2.98·24-s − 3.77·25-s + 0.454·26-s − 27-s − 0.388·28-s + ⋯ |
| L(s) = 1 | + 0.930·2-s − 0.577·3-s − 0.133·4-s − 0.495·5-s − 0.537·6-s + 0.549·7-s − 1.05·8-s + 0.333·9-s − 0.461·10-s − 0.252·11-s + 0.0771·12-s + 0.0957·13-s + 0.511·14-s + 0.286·15-s − 0.848·16-s + 1.23·17-s + 0.310·18-s + 1.26·19-s + 0.0662·20-s − 0.317·21-s − 0.234·22-s − 0.248·23-s + 0.609·24-s − 0.754·25-s + 0.0891·26-s − 0.192·27-s − 0.0734·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 677 | \( 1 + T \) |
| good | 2 | \( 1 - 1.31T + 2T^{2} \) |
| 5 | \( 1 + 1.10T + 5T^{2} \) |
| 7 | \( 1 - 1.45T + 7T^{2} \) |
| 11 | \( 1 + 0.837T + 11T^{2} \) |
| 13 | \( 1 - 0.345T + 13T^{2} \) |
| 17 | \( 1 - 5.10T + 17T^{2} \) |
| 19 | \( 1 - 5.49T + 19T^{2} \) |
| 23 | \( 1 + 1.19T + 23T^{2} \) |
| 29 | \( 1 + 8.98T + 29T^{2} \) |
| 31 | \( 1 + 5.20T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 5.55T + 41T^{2} \) |
| 43 | \( 1 + 5.69T + 43T^{2} \) |
| 47 | \( 1 + 7.62T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 8.37T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 + 3.48T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 5.59T + 79T^{2} \) |
| 83 | \( 1 - 5.09T + 83T^{2} \) |
| 89 | \( 1 + 2.22T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.740557217502434325151433606708, −7.73481129018249922841399499897, −7.25820826469164465531872588672, −5.93775845532121430106839054862, −5.46460570650668121374152265103, −4.81712327348275167778247460018, −3.80152926649393509592155530451, −3.24878598301584658142093342209, −1.61347681306554310077939031531, 0,
1.61347681306554310077939031531, 3.24878598301584658142093342209, 3.80152926649393509592155530451, 4.81712327348275167778247460018, 5.46460570650668121374152265103, 5.93775845532121430106839054862, 7.25820826469164465531872588672, 7.73481129018249922841399499897, 8.740557217502434325151433606708
