L(s) = 1 | + (−0.604 + 0.161i)2-s + (−1.39 + 0.804i)4-s + (−0.965 + 0.965i)5-s + (2.12 + 1.57i)7-s + (1.59 − 1.59i)8-s + (0.427 − 0.739i)10-s + (−1.26 − 4.72i)11-s + (1.35 − 3.34i)13-s + (−1.53 − 0.610i)14-s + (0.902 − 1.56i)16-s + (2.72 + 4.72i)17-s + (4.47 + 1.19i)19-s + (0.568 − 2.12i)20-s + (1.53 + 2.65i)22-s + (3.14 + 1.81i)23-s + ⋯ |
L(s) = 1 | + (−0.427 + 0.114i)2-s + (−0.696 + 0.402i)4-s + (−0.431 + 0.431i)5-s + (0.802 + 0.596i)7-s + (0.564 − 0.564i)8-s + (0.135 − 0.233i)10-s + (−0.382 − 1.42i)11-s + (0.374 − 0.927i)13-s + (−0.411 − 0.163i)14-s + (0.225 − 0.390i)16-s + (0.660 + 1.14i)17-s + (1.02 + 0.275i)19-s + (0.127 − 0.474i)20-s + (0.326 + 0.565i)22-s + (0.655 + 0.378i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.873942 + 0.533647i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.873942 + 0.533647i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.12 - 1.57i)T \) |
| 13 | \( 1 + (-1.35 + 3.34i)T \) |
good | 2 | \( 1 + (0.604 - 0.161i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (0.965 - 0.965i)T - 5iT^{2} \) |
| 11 | \( 1 + (1.26 + 4.72i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.72 - 4.72i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.47 - 1.19i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.14 - 1.81i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.00 - 1.74i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.91 - 5.91i)T - 31iT^{2} \) |
| 37 | \( 1 + (2.84 + 10.6i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.08 - 4.04i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (0.669 - 0.386i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \) |
| 53 | \( 1 - 6.72T + 53T^{2} \) |
| 59 | \( 1 + (3.87 - 14.4i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-0.210 + 0.121i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.55 + 1.48i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.711 + 2.65i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-2.17 - 2.17i)T + 73iT^{2} \) |
| 79 | \( 1 - 8.38T + 79T^{2} \) |
| 83 | \( 1 + (-11.2 + 11.2i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.25 - 0.872i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-9.46 - 2.53i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.65331052602093182378243056213, −9.236497845352846960109590508965, −8.656185191133257519414508317955, −7.87223450577522617200908724101, −7.40789736176911564881759294284, −5.74064860880074074739591015148, −5.28978127299323903129415945624, −3.72747518453160245544827394473, −3.15582014898764511755091660158, −1.13667068597878169235421662330,
0.77548274794261443281656269418, 2.02878450519809001392978018445, 3.91967483069423216943328471086, 4.78613934071016998484685325897, 5.24057300393381162473831171259, 6.94600810803820177757567301432, 7.64213933516907625705481995976, 8.417039650724850232234080168634, 9.403162826000980113846099508065, 9.867759774358556900425622735786