L(s) = 1 | + (33.0 − 33.0i)3-s + (2.96e3 + 2.96e3i)7-s − 2.18e3i·9-s + 2.70e4·11-s + (1.63e4 − 1.63e4i)13-s + (5.21e4 + 5.21e4i)17-s + 9.93e4i·19-s + 1.95e5·21-s + (2.52e5 − 2.52e5i)23-s + (−7.23e4 − 7.23e4i)27-s + 1.83e5i·29-s + 3.87e5·31-s + (8.93e5 − 8.93e5i)33-s + (−1.68e6 − 1.68e6i)37-s − 1.08e6i·39-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (1.23 + 1.23i)7-s − 0.333i·9-s + 1.84·11-s + (0.573 − 0.573i)13-s + (0.624 + 0.624i)17-s + 0.762i·19-s + 1.00·21-s + (0.902 − 0.902i)23-s + (−0.136 − 0.136i)27-s + 0.259i·29-s + 0.420·31-s + (0.753 − 0.753i)33-s + (−0.899 − 0.899i)37-s − 0.468i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(4.063287535\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.063287535\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-33.0 + 33.0i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-2.96e3 - 2.96e3i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 - 2.70e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (-1.63e4 + 1.63e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (-5.21e4 - 5.21e4i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 - 9.93e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (-2.52e5 + 2.52e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 - 1.83e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 3.87e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + (1.68e6 + 1.68e6i)T + 3.51e12iT^{2} \) |
| 41 | \( 1 - 1.14e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (2.09e6 - 2.09e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + (2.73e4 + 2.73e4i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + (3.82e6 - 3.82e6i)T - 6.22e13iT^{2} \) |
| 59 | \( 1 - 1.30e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.59e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + (1.20e7 + 1.20e7i)T + 4.06e14iT^{2} \) |
| 71 | \( 1 + 3.95e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (2.66e7 - 2.66e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 + 7.09e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (-1.06e7 + 1.06e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 - 4.35e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (5.13e6 + 5.13e6i)T + 7.83e15iT^{2} \) |
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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
−10.42939599452702450111086769266, −8.997997400563770934073019772650, −8.649991803602470304580160525526, −7.69048724102959854014823510184, −6.40782698054831777960807939713, −5.59315530323118990341193516812, −4.29155192723700205847231917118, −3.11289960522390672742062189748, −1.77324047619569829880648011119, −1.14814352762267615776199555666,
0.952348276526806160289634325284, 1.63012541789907490049798585908, 3.39760461953417491560705904350, 4.20774002872111435756800659359, 5.04891215028663773948239071534, 6.66104036973691550822585008439, 7.40601454151955177369013553291, 8.532961683703872248417147692892, 9.300403685746531171895285103614, 10.28426297580861748892635213354