| L(s) = 1 | + 2.40·2-s − 1.49·3-s + 3.79·4-s − 5-s − 3.59·6-s − 3.00·7-s + 4.33·8-s − 0.765·9-s − 2.40·10-s + 11-s − 5.67·12-s + 6.33·13-s − 7.23·14-s + 1.49·15-s + 2.83·16-s − 3.35·17-s − 1.84·18-s + 5.48·19-s − 3.79·20-s + 4.48·21-s + 2.40·22-s − 0.251·23-s − 6.47·24-s + 25-s + 15.2·26-s + 5.62·27-s − 11.4·28-s + ⋯ |
| L(s) = 1 | + 1.70·2-s − 0.863·3-s + 1.89·4-s − 0.447·5-s − 1.46·6-s − 1.13·7-s + 1.53·8-s − 0.255·9-s − 0.761·10-s + 0.301·11-s − 1.63·12-s + 1.75·13-s − 1.93·14-s + 0.385·15-s + 0.707·16-s − 0.813·17-s − 0.434·18-s + 1.25·19-s − 0.849·20-s + 0.979·21-s + 0.513·22-s − 0.0524·23-s − 1.32·24-s + 0.200·25-s + 2.99·26-s + 1.08·27-s − 2.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 3 | \( 1 + 1.49T + 3T^{2} \) |
| 7 | \( 1 + 3.00T + 7T^{2} \) |
| 13 | \( 1 - 6.33T + 13T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 - 5.48T + 19T^{2} \) |
| 23 | \( 1 + 0.251T + 23T^{2} \) |
| 29 | \( 1 + 7.82T + 29T^{2} \) |
| 31 | \( 1 + 5.08T + 31T^{2} \) |
| 37 | \( 1 + 5.97T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 - 2.22T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 6.59T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 2.33T + 61T^{2} \) |
| 67 | \( 1 + 0.0824T + 67T^{2} \) |
| 71 | \( 1 + 0.634T + 71T^{2} \) |
| 79 | \( 1 - 2.14T + 79T^{2} \) |
| 83 | \( 1 - 1.68T + 83T^{2} \) |
| 89 | \( 1 - 4.23T + 89T^{2} \) |
| 97 | \( 1 - 2.98T + 97T^{2} \) |
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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
−7.76420020427668009614880398736, −6.71471480270060099270104183181, −6.48481230776816259133663919872, −5.71589252599710585308808456776, −5.23582312371079906909288517091, −4.22560488313288671211486548077, −3.45226525985149685077203392414, −3.13713764918118530310878371546, −1.61826857984527504154772815448, 0,
1.61826857984527504154772815448, 3.13713764918118530310878371546, 3.45226525985149685077203392414, 4.22560488313288671211486548077, 5.23582312371079906909288517091, 5.71589252599710585308808456776, 6.48481230776816259133663919872, 6.71471480270060099270104183181, 7.76420020427668009614880398736
