L(s) = 1 | + (−0.531 + 2.95i)3-s + (−3.31 − 5.73i)5-s + (−8.46 − 4.88i)7-s + (−8.43 − 3.13i)9-s + (10.0 + 5.80i)11-s + (−8.93 − 15.4i)13-s + (18.6 − 6.72i)15-s + 2.37·17-s − 14.0i·19-s + (18.9 − 22.4i)21-s + (−35.9 + 20.7i)23-s + (−9.43 + 16.3i)25-s + (13.7 − 23.2i)27-s + (−5.68 + 9.85i)29-s + (−18.0 + 10.4i)31-s + ⋯ |
L(s) = 1 | + (−0.177 + 0.984i)3-s + (−0.662 − 1.14i)5-s + (−1.20 − 0.698i)7-s + (−0.937 − 0.348i)9-s + (0.914 + 0.528i)11-s + (−0.687 − 1.19i)13-s + (1.24 − 0.448i)15-s + 0.139·17-s − 0.738i·19-s + (0.901 − 1.06i)21-s + (−1.56 + 0.902i)23-s + (−0.377 + 0.653i)25-s + (0.509 − 0.860i)27-s + (−0.196 + 0.339i)29-s + (−0.581 + 0.335i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.235749 - 0.411691i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.235749 - 0.411691i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.531 - 2.95i)T \) |
good | 5 | \( 1 + (3.31 + 5.73i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (8.46 + 4.88i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-10.0 - 5.80i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (8.93 + 15.4i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 2.37T + 289T^{2} \) |
| 19 | \( 1 + 14.0iT - 361T^{2} \) |
| 23 | \( 1 + (35.9 - 20.7i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (5.68 - 9.85i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (18.0 - 10.4i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 35.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (2.62 + 4.55i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (54.4 + 31.4i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-4.28 - 2.47i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 75.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-50.3 + 29.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-5.93 + 10.2i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (20.6 - 11.9i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 46.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 104.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (18.0 + 10.4i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (24.4 + 14.0i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 73.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-71.1 + 123. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−12.43127962450953390065550302025, −11.63511778662182450495543800090, −10.18654063696631441463314295852, −9.609873684964610761804736076221, −8.537632165002052997591871965450, −7.19512027169782394230917059294, −5.64894841108081471784004401659, −4.41124153209703490763362872378, −3.48215932430672087352252771053, −0.30293434688714675451298372638,
2.40246481495398338765007024614, 3.74126239378080872738435778174, 6.10667517583479517604233438262, 6.60550062212783015418969068058, 7.68896598218475412070650247835, 8.984286733099242704991860804894, 10.19064571976203390154974565882, 11.73745760694300708164690877841, 11.84622623174032605864329712243, 13.09931676970501617814387300834