| L(s) = 1 | + (1 − i)2-s + 14i·4-s + (−20 + 15i)5-s + (−26 + 26i)7-s + (30 + 30i)8-s + (−5 + 35i)10-s + 8·11-s + (139 + 139i)13-s + 52i·14-s − 164·16-s + (1 − i)17-s − 180i·19-s + (−210 − 280i)20-s + (8 − 8i)22-s + (166 + 166i)23-s + ⋯ |
| L(s) = 1 | + (0.250 − 0.250i)2-s + 0.875i·4-s + (−0.800 + 0.599i)5-s + (−0.530 + 0.530i)7-s + (0.468 + 0.468i)8-s + (−0.0500 + 0.350i)10-s + 0.0661·11-s + (0.822 + 0.822i)13-s + 0.265i·14-s − 0.640·16-s + (0.00346 − 0.00346i)17-s − 0.498i·19-s + (−0.525 − 0.700i)20-s + (0.0165 − 0.0165i)22-s + (0.313 + 0.313i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0898 - 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0898 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.859358 + 0.940333i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.859358 + 0.940333i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (20 - 15i)T \) |
| good | 2 | \( 1 + (-1 + i)T - 16iT^{2} \) |
| 7 | \( 1 + (26 - 26i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 - 8T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-139 - 139i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + 180iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (-166 - 166i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + 480iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 572T + 9.23e5T^{2} \) |
| 37 | \( 1 + (251 - 251i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 1.68e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-1.47e3 - 1.47e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (2.47e3 - 2.47e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-3.33e3 - 3.33e3i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 + 3.66e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.59e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-874 + 874i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 6.06e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (791 + 791i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 + 9.12e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (5.65e3 + 5.65e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 - 2.16e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (6.55e3 - 6.55e3i)T - 8.85e7iT^{2} \) |
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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
−15.55222894225803555802467798423, −14.10967538830470540286367616798, −12.91969857715237979330609905476, −11.82650786254934343262423732323, −11.03133743598720098513824575313, −9.142287699976660856558823307310, −7.85537123137379967680051847125, −6.51732994403854168367749072818, −4.20996161327773642403050188994, −2.87320781414916755354993937399,
0.78864456614154069033762532842, 3.89685859819881376462260890021, 5.45821660175173617123785637123, 6.95794904916485022181441899410, 8.498441263019147602559639010236, 10.00740106028953358180269250789, 11.10623609502285657633247387279, 12.64275896354794503897853033770, 13.62289884611926865718993429014, 14.92007943921848102007623341243
