| L(s) = 1 | − 3·3-s + (−6.39 − 3.69i)5-s + (−3.39 − 5.88i)7-s + 9·9-s + (5.29 − 3.05i)11-s + (8.39 − 14.5i)13-s + (19.1 + 11.0i)15-s − 25.1i·17-s − 17.5·19-s + (10.1 + 17.6i)21-s + (−12.3 − 7.15i)23-s + (14.7 + 25.6i)25-s − 27·27-s + (−16.1 + 9.35i)29-s + (−23.3 + 40.5i)31-s + ⋯ |
| L(s) = 1 | − 3-s + (−1.27 − 0.738i)5-s + (−0.485 − 0.841i)7-s + 9-s + (0.481 − 0.278i)11-s + (0.646 − 1.11i)13-s + (1.27 + 0.738i)15-s − 1.48i·17-s − 0.926·19-s + (0.485 + 0.841i)21-s + (−0.539 − 0.311i)23-s + (0.591 + 1.02i)25-s − 27-s + (−0.558 + 0.322i)29-s + (−0.754 + 1.30i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2321524320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2321524320\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 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
−10.16765645829244065422338583918, −9.076513948006652075988417868002, −8.030667746807309788959209014442, −7.25040704541379506212981675515, −6.34584305542696086593599559937, −5.17320620845654811577059928492, −4.27325879332092720008578462445, −3.44296817418871692739443809990, −0.971262284789838333069819944982, −0.13172050777226013689183905538,
1.94701430534579390970176225435, 3.78551219304738798781646494447, 4.23563988341693451252665424927, 5.91927850587679058184439227326, 6.41293590922224388091077974071, 7.34960226281422822109811851782, 8.362834155522881114019828123137, 9.408147805155981254307733046856, 10.42463789882155626792670482796, 11.37044836152985448410250414595
