| L(s) = 1 | + 2-s − 3-s + 4-s + 2.77·5-s − 6-s − 3.71·7-s + 8-s + 9-s + 2.77·10-s + 2.24·11-s − 12-s − 5.30·13-s − 3.71·14-s − 2.77·15-s + 16-s − 2.77·17-s + 18-s − 1.83·19-s + 2.77·20-s + 3.71·21-s + 2.24·22-s − 23-s − 24-s + 2.71·25-s − 5.30·26-s − 27-s − 3.71·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.24·5-s − 0.408·6-s − 1.40·7-s + 0.353·8-s + 0.333·9-s + 0.878·10-s + 0.678·11-s − 0.288·12-s − 1.47·13-s − 0.994·14-s − 0.717·15-s + 0.250·16-s − 0.673·17-s + 0.235·18-s − 0.421·19-s + 0.621·20-s + 0.811·21-s + 0.479·22-s − 0.208·23-s − 0.204·24-s + 0.543·25-s − 1.04·26-s − 0.192·27-s − 0.702·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 2.77T + 5T^{2} \) |
| 7 | \( 1 + 3.71T + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 17 | \( 1 + 2.77T + 17T^{2} \) |
| 19 | \( 1 + 1.83T + 19T^{2} \) |
| 31 | \( 1 - 7.80T + 31T^{2} \) |
| 37 | \( 1 + 4.52T + 37T^{2} \) |
| 41 | \( 1 + 3.47T + 41T^{2} \) |
| 43 | \( 1 + 4.77T + 43T^{2} \) |
| 47 | \( 1 + 1.33T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 3.50T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 5.92T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 + 4.94T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.906783495673297864829469949649, −6.79115273217686172828324758042, −6.51249639278645606868290197078, −6.02282587228024304845954778448, −5.06309621831789021477706448453, −4.50330369119786009672423401738, −3.35727265038420938171564883331, −2.55299667323863577694166823675, −1.65573525786387325270004352873, 0,
1.65573525786387325270004352873, 2.55299667323863577694166823675, 3.35727265038420938171564883331, 4.50330369119786009672423401738, 5.06309621831789021477706448453, 6.02282587228024304845954778448, 6.51249639278645606868290197078, 6.79115273217686172828324758042, 7.906783495673297864829469949649
