| L(s) = 1 | + (−5.77 + 7.94i)2-s + (−24.0 + 7.80i)3-s + (−19.9 − 61.3i)4-s + (11.5 + 54.6i)5-s + (76.6 − 235. i)6-s + 104. i·7-s + (304. + 98.8i)8-s + (319. − 231. i)9-s + (−501. − 224. i)10-s + (37.6 + 27.3i)11-s + (958. + 1.31e3i)12-s + (−609. − 839. i)13-s + (−832. − 604. i)14-s + (−703. − 1.22e3i)15-s + (−871. + 633. i)16-s + (640. + 208. i)17-s + ⋯ |
| L(s) = 1 | + (−1.02 + 1.40i)2-s + (−1.54 + 0.500i)3-s + (−0.623 − 1.91i)4-s + (0.206 + 0.978i)5-s + (0.869 − 2.67i)6-s + 0.807i·7-s + (1.68 + 0.546i)8-s + (1.31 − 0.954i)9-s + (−1.58 − 0.708i)10-s + (0.0937 + 0.0681i)11-s + (1.92 + 2.64i)12-s + (−1.00 − 1.37i)13-s + (−1.13 − 0.824i)14-s + (−0.807 − 1.40i)15-s + (−0.851 + 0.618i)16-s + (0.537 + 0.174i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0856719 - 0.0449538i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0856719 - 0.0449538i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-11.5 - 54.6i)T \) |
| good | 2 | \( 1 + (5.77 - 7.94i)T + (-9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (24.0 - 7.80i)T + (196. - 142. i)T^{2} \) |
| 7 | \( 1 - 104. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-37.6 - 27.3i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (609. + 839. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-640. - 208. i)T + (1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (750. - 2.30e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-474. + 652. i)T + (-1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (1.48e3 + 4.57e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-724. + 2.22e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (4.99e3 + 6.87e3i)T + (-2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (2.93e3 - 2.13e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 3.64e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (1.63e4 - 5.31e3i)T + (1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-1.48e4 + 4.81e3i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.65e4 - 1.20e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (2.39e4 + 1.74e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (4.35e4 + 1.41e4i)T + (1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-1.17e3 - 3.60e3i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (1.11e4 - 1.53e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-2.65e3 - 8.17e3i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (5.33e3 + 1.73e3i)T + (3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-8.08e4 - 5.87e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (7.66e4 - 2.48e4i)T + (6.94e9 - 5.04e9i)T^{2} \) |
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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
−17.42683747313839789250962245288, −16.59832898077415172952134220238, −15.36675352720857641988933794983, −14.74603366889272291890373535509, −12.18746153117945828270742661795, −10.53991135622393203544285119339, −9.807949492855477597753278015155, −7.75353408290058919518258423157, −6.21445030305255718733564594147, −5.48328079961480468531830831423,
0.11693111545390926646327254248, 1.43849454018043758537234578047, 4.73554291571208065586322107023, 7.10250575617054596148949028331, 9.087985272375040193583528955116, 10.38852339555584376956737967782, 11.56210156376355212964976041372, 12.26903725582179635948977942378, 13.39609783847927162502921533398, 16.59985723717321906482169616968
