L(s) = 1 | + (−0.619 + 2.31i)2-s + (−1.28 + 1.16i)3-s + (−3.23 − 1.86i)4-s + (−1.69 − 1.69i)5-s + (−1.89 − 3.68i)6-s + (1.36 − 0.366i)7-s + (2.93 − 2.93i)8-s + (0.292 − 2.98i)9-s + (4.96 − 2.86i)10-s + (1.69 + 0.453i)11-s + (6.31 − 1.36i)12-s + 3.38i·14-s + (4.14 + 0.202i)15-s + (1.23 + 2.13i)16-s + (1.07 − 1.85i)17-s + (6.72 + 2.52i)18-s + ⋯ |
L(s) = 1 | + (−0.438 + 1.63i)2-s + (−0.740 + 0.671i)3-s + (−1.61 − 0.933i)4-s + (−0.757 − 0.757i)5-s + (−0.773 − 1.50i)6-s + (0.516 − 0.138i)7-s + (1.03 − 1.03i)8-s + (0.0975 − 0.995i)9-s + (1.56 − 0.906i)10-s + (0.510 + 0.136i)11-s + (1.82 − 0.394i)12-s + 0.904i·14-s + (1.06 + 0.0523i)15-s + (0.308 + 0.533i)16-s + (0.260 − 0.450i)17-s + (1.58 + 0.595i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.532046 + 0.403935i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.532046 + 0.403935i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.28 - 1.16i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.619 - 2.31i)T + (-1.73 - i)T^{2} \) |
| 5 | \( 1 + (1.69 + 1.69i)T + 5iT^{2} \) |
| 7 | \( 1 + (-1.36 + 0.366i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.69 - 0.453i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.07 + 1.85i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.267 - i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.79 + 2.76i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.46 - 4.46i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.76 + 6.59i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.166 + 0.619i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-7.09 - 4.09i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.77 + 6.77i)T - 47iT^{2} \) |
| 53 | \( 1 + 4.62iT - 53T^{2} \) |
| 59 | \( 1 + (1.23 + 4.62i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.46 - 2.26i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (4.62 - 1.23i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-6.09 - 6.09i)T + 73iT^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + (-1.23 - 1.23i)T + 83iT^{2} \) |
| 89 | \( 1 + (-9.70 - 2.60i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (3.36 + 12.5i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−11.02799179097286943065414696313, −9.864321376018370397016787287596, −9.086848917704843004949066397024, −8.328670341306335446662152313018, −7.46019663829468096226655595068, −6.54363033783400013894503016156, −5.51067410497168262336229020497, −4.77898448597681675133552923683, −3.96024065086321508824707359009, −0.67182908395709811945723892526,
1.07502410790186060628515781433, 2.37419873775286964796565961732, 3.58302209362820257260827009076, 4.68028899608053317726174270188, 6.13043879799524928349476195560, 7.29852028586964167182892325580, 8.130242298340361234306026746810, 9.132131137258583561502246019587, 10.33911899773113299386503566389, 10.92854689729093374876693264787