| L(s) = 1 | − 2.00·2-s − 0.510·3-s + 2.00·4-s + 2.33·5-s + 1.02·6-s − 3.93·7-s − 0.00949·8-s − 2.73·9-s − 4.67·10-s + 11-s − 1.02·12-s + 5.23·13-s + 7.87·14-s − 1.19·15-s − 3.99·16-s + 17-s + 5.48·18-s − 2.97·19-s + 4.68·20-s + 2.00·21-s − 2.00·22-s + 0.0593·23-s + 0.00484·24-s + 0.453·25-s − 10.4·26-s + 2.92·27-s − 7.89·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.294·3-s + 1.00·4-s + 1.04·5-s + 0.416·6-s − 1.48·7-s − 0.00335·8-s − 0.913·9-s − 1.47·10-s + 0.301·11-s − 0.295·12-s + 1.45·13-s + 2.10·14-s − 0.307·15-s − 0.997·16-s + 0.242·17-s + 1.29·18-s − 0.681·19-s + 1.04·20-s + 0.438·21-s − 0.426·22-s + 0.0123·23-s + 0.000988·24-s + 0.0907·25-s − 2.05·26-s + 0.563·27-s − 1.49·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5728501294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5728501294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2.00T + 2T^{2} \) |
| 3 | \( 1 + 0.510T + 3T^{2} \) |
| 5 | \( 1 - 2.33T + 5T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 19 | \( 1 + 2.97T + 19T^{2} \) |
| 23 | \( 1 - 0.0593T + 23T^{2} \) |
| 29 | \( 1 + 9.76T + 29T^{2} \) |
| 31 | \( 1 + 0.151T + 31T^{2} \) |
| 37 | \( 1 + 1.80T + 37T^{2} \) |
| 41 | \( 1 - 7.48T + 41T^{2} \) |
| 47 | \( 1 + 1.63T + 47T^{2} \) |
| 53 | \( 1 - 2.27T + 53T^{2} \) |
| 59 | \( 1 - 4.09T + 59T^{2} \) |
| 61 | \( 1 + 7.28T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 9.00T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 5.94T + 83T^{2} \) |
| 89 | \( 1 + 9.06T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.000894679335042298333721520336, −7.15448422936205702833219283522, −6.39051341373403598150705146559, −6.03524159145875163476621996118, −5.48396682164946860974994330646, −4.11341512700105288995066807933, −3.31679867115320204473562232704, −2.38554314095599476175452161558, −1.53515623157956248432681814452, −0.48024371607492637427717879563,
0.48024371607492637427717879563, 1.53515623157956248432681814452, 2.38554314095599476175452161558, 3.31679867115320204473562232704, 4.11341512700105288995066807933, 5.48396682164946860974994330646, 6.03524159145875163476621996118, 6.39051341373403598150705146559, 7.15448422936205702833219283522, 8.000894679335042298333721520336
