L(s) = 1 | − 2.41·2-s + 1.23·3-s + 3.82·4-s + 5-s − 2.98·6-s − 1.94·7-s − 4.39·8-s − 1.46·9-s − 2.41·10-s − 11-s + 4.73·12-s + 0.413·13-s + 4.68·14-s + 1.23·15-s + 2.96·16-s + 3.32·17-s + 3.54·18-s + 4.62·19-s + 3.82·20-s − 2.40·21-s + 2.41·22-s − 4.22·23-s − 5.44·24-s + 25-s − 0.997·26-s − 5.53·27-s − 7.41·28-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 0.714·3-s + 1.91·4-s + 0.447·5-s − 1.21·6-s − 0.733·7-s − 1.55·8-s − 0.489·9-s − 0.763·10-s − 0.301·11-s + 1.36·12-s + 0.114·13-s + 1.25·14-s + 0.319·15-s + 0.741·16-s + 0.805·17-s + 0.834·18-s + 1.06·19-s + 0.854·20-s − 0.524·21-s + 0.514·22-s − 0.881·23-s − 1.11·24-s + 0.200·25-s − 0.195·26-s − 1.06·27-s − 1.40·28-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 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 - 1.23T + 3T^{2} \) |
| 7 | \( 1 + 1.94T + 7T^{2} \) |
| 13 | \( 1 - 0.413T + 13T^{2} \) |
| 17 | \( 1 - 3.32T + 17T^{2} \) |
| 19 | \( 1 - 4.62T + 19T^{2} \) |
| 23 | \( 1 + 4.22T + 23T^{2} \) |
| 29 | \( 1 + 7.63T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 - 3.42T + 37T^{2} \) |
| 41 | \( 1 - 5.48T + 41T^{2} \) |
| 43 | \( 1 - 7.64T + 43T^{2} \) |
| 47 | \( 1 - 3.44T + 47T^{2} \) |
| 53 | \( 1 + 2.55T + 53T^{2} \) |
| 59 | \( 1 + 0.900T + 59T^{2} \) |
| 61 | \( 1 - 7.14T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 1.57T + 71T^{2} \) |
| 79 | \( 1 + 2.55T + 79T^{2} \) |
| 83 | \( 1 - 9.24T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 6.25T + 97T^{2} \) |
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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
−7.993031629654505243751743335998, −7.77529712227819878597449950837, −6.98130386041573630026928973146, −5.99366980871710193513754131300, −5.51377108427236183538461990462, −3.89216545128173713792510475351, −2.97225058522033083539149137783, −2.32901419518649937991707422582, −1.28448443334658737831525885569, 0,
1.28448443334658737831525885569, 2.32901419518649937991707422582, 2.97225058522033083539149137783, 3.89216545128173713792510475351, 5.51377108427236183538461990462, 5.99366980871710193513754131300, 6.98130386041573630026928973146, 7.77529712227819878597449950837, 7.993031629654505243751743335998