L(s) = 1 | + (6.39 − 3.69i)5-s + (3.39 − 5.88i)7-s + (5.29 + 3.05i)11-s + (8.39 + 14.5i)13-s − 25.1i·17-s + 17.5·19-s + (−12.3 + 7.15i)23-s + (14.7 − 25.6i)25-s + (16.1 + 9.35i)29-s + (23.3 + 40.5i)31-s − 50.2i·35-s + 49.5·37-s + (34.5 − 19.9i)41-s + (−22.0 + 38.2i)43-s + (−28.8 − 16.6i)47-s + ⋯ |
L(s) = 1 | + (1.27 − 0.738i)5-s + (0.485 − 0.841i)7-s + (0.481 + 0.278i)11-s + (0.646 + 1.11i)13-s − 1.48i·17-s + 0.926·19-s + (−0.539 + 0.311i)23-s + (0.591 − 1.02i)25-s + (0.558 + 0.322i)29-s + (0.754 + 1.30i)31-s − 1.43i·35-s + 1.34·37-s + (0.841 − 0.485i)41-s + (−0.513 + 0.890i)43-s + (−0.612 − 0.353i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.210221224\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.210221224\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-6.39 + 3.69i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-3.39 + 5.88i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-5.29 - 3.05i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-8.39 - 14.5i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 25.1iT - 289T^{2} \) |
| 19 | \( 1 - 17.5T + 361T^{2} \) |
| 23 | \( 1 + (12.3 - 7.15i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-16.1 - 9.35i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-23.3 - 40.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 49.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-34.5 + 19.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (22.0 - 38.2i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (28.8 + 16.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 10.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (14.2 - 8.25i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-10.6 + 18.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (43.4 + 75.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 30.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 48.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + (55.7 - 96.6i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (85.0 + 49.1i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 75.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-70.2 + 121. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.289835558728524985685840245350, −8.390386356475046283667421485084, −7.36286205918262603513109287545, −6.65526314586160661621725148814, −5.78859510994078166385068493615, −4.84259040291651436851628870467, −4.31513195132048642340747444130, −2.93799035521976640228784725560, −1.62440547603990938748605767666, −1.04086658623673939089848364594,
1.18231373429416668209257575379, 2.24443217292774221977887144493, 3.03054441190722984673246068919, 4.19344069048075078939807105703, 5.54481195044691104813458150796, 5.94694775782483659683721295436, 6.49390548900001951940319724759, 7.84602891344571365659115542027, 8.377027935342915330221197839656, 9.310723029435831931838523764845