| L(s) = 1 | − 3.18·3-s + 2.23·5-s + 2.57·7-s + 7.16·9-s + 2.62·11-s − 2.46·13-s − 7.13·15-s − 1.10·17-s + 5.55·19-s − 8.20·21-s + 1.46·23-s + 0.0107·25-s − 13.2·27-s − 6.66·29-s + 10.6·31-s − 8.38·33-s + 5.76·35-s + 9.34·37-s + 7.85·39-s + 2.42·41-s + 8.37·43-s + 16.0·45-s + 2.50·47-s − 0.371·49-s + 3.51·51-s + 9.61·53-s + 5.88·55-s + ⋯ |
| L(s) = 1 | − 1.84·3-s + 1.00·5-s + 0.973·7-s + 2.38·9-s + 0.792·11-s − 0.683·13-s − 1.84·15-s − 0.267·17-s + 1.27·19-s − 1.79·21-s + 0.305·23-s + 0.00214·25-s − 2.55·27-s − 1.23·29-s + 1.90·31-s − 1.45·33-s + 0.974·35-s + 1.53·37-s + 1.25·39-s + 0.379·41-s + 1.27·43-s + 2.39·45-s + 0.366·47-s − 0.0531·49-s + 0.491·51-s + 1.32·53-s + 0.793·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.679424194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.679424194\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 3.18T + 3T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 - 2.57T + 7T^{2} \) |
| 11 | \( 1 - 2.62T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 + 1.10T + 17T^{2} \) |
| 19 | \( 1 - 5.55T + 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 + 6.66T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 - 9.34T + 37T^{2} \) |
| 41 | \( 1 - 2.42T + 41T^{2} \) |
| 43 | \( 1 - 8.37T + 43T^{2} \) |
| 47 | \( 1 - 2.50T + 47T^{2} \) |
| 53 | \( 1 - 9.61T + 53T^{2} \) |
| 59 | \( 1 + 3.69T + 59T^{2} \) |
| 61 | \( 1 + 8.77T + 61T^{2} \) |
| 67 | \( 1 - 4.18T + 67T^{2} \) |
| 71 | \( 1 + 5.99T + 71T^{2} \) |
| 73 | \( 1 + 6.74T + 73T^{2} \) |
| 79 | \( 1 - 2.05T + 79T^{2} \) |
| 83 | \( 1 - 4.68T + 83T^{2} \) |
| 89 | \( 1 - 4.75T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−7.70357178498921148172209963123, −7.32632869984396447245871194067, −6.33388097652166127736129982746, −5.97720300896848559128445304324, −5.27256266536114463783218336100, −4.72367790719013755225512767101, −4.05875984767399493719060521369, −2.51745199600344626379569654783, −1.49798790949922000587028742705, −0.837649459853978641483003453566,
0.837649459853978641483003453566, 1.49798790949922000587028742705, 2.51745199600344626379569654783, 4.05875984767399493719060521369, 4.72367790719013755225512767101, 5.27256266536114463783218336100, 5.97720300896848559128445304324, 6.33388097652166127736129982746, 7.32632869984396447245871194067, 7.70357178498921148172209963123
