| L(s) = 1 | + (5.68 + 14.9i)2-s + (68.3 + 22.1i)3-s + (−191. + 169. i)4-s + (454. + 429. i)5-s + (56.2 + 1.14e3i)6-s + 1.40e3i·7-s + (−3.63e3 − 1.89e3i)8-s + (−1.13e3 − 824. i)9-s + (−3.83e3 + 9.23e3i)10-s + (−9.62e3 − 1.32e4i)11-s + (−1.68e4 + 7.36e3i)12-s + (−4.21e4 − 3.06e4i)13-s + (−2.10e4 + 8.00e3i)14-s + (2.15e4 + 3.93e4i)15-s + (7.73e3 − 6.50e4i)16-s + (1.69e4 + 5.21e4i)17-s + ⋯ |
| L(s) = 1 | + (0.355 + 0.934i)2-s + (0.843 + 0.273i)3-s + (−0.747 + 0.664i)4-s + (0.727 + 0.686i)5-s + (0.0433 + 0.885i)6-s + 0.586i·7-s + (−0.886 − 0.463i)8-s + (−0.172 − 0.125i)9-s + (−0.383 + 0.923i)10-s + (−0.657 − 0.905i)11-s + (−0.812 + 0.355i)12-s + (−1.47 − 1.07i)13-s + (−0.548 + 0.208i)14-s + (0.425 + 0.778i)15-s + (0.118 − 0.993i)16-s + (0.202 + 0.624i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.527046 - 0.751133i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.527046 - 0.751133i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-5.68 - 14.9i)T \) |
| 5 | \( 1 + (-454. - 429. i)T \) |
| good | 3 | \( 1 + (-68.3 - 22.1i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 - 1.40e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (9.62e3 + 1.32e4i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (4.21e4 + 3.06e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-1.69e4 - 5.21e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (1.04e5 - 3.39e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (-5.88e4 - 8.09e4i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (2.98e5 - 9.18e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-1.09e5 + 3.56e4i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (1.45e6 + 1.05e6i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (7.99e5 + 5.81e5i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 - 5.50e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-4.04e5 - 1.31e5i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (3.15e6 - 9.71e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (-4.58e5 + 6.31e5i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-1.10e7 + 8.04e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (5.96e6 - 1.93e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (3.74e7 + 1.21e7i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (-3.62e7 + 2.63e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (-2.26e7 - 7.35e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (8.29e7 - 2.69e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-4.74e7 + 3.44e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (3.24e7 - 9.98e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−13.28035626952042742568486454784, −12.37764983707979757837287555987, −10.61607144168094321854983248751, −9.484482995133099009030864399336, −8.520256011754127557884933066798, −7.54239950034219420085587880998, −6.08733328300977531766731847482, −5.25557426673230734943974529188, −3.36448185531515261939143937215, −2.58592501108835279204892744554,
0.18117305125616805798944777922, 1.91340094149355431821869181473, 2.50768426350985223299197991363, 4.33556572975432609138049543997, 5.24355521628151269277279745387, 7.10567988291211740137419881383, 8.526897643928899319193648047545, 9.551202741775361356066603329414, 10.24014457706362803001558316720, 11.75744285577788644490748110824
