| L(s) = 1 | + (3.57 − 2.06i)2-s + (−32.2 + 55.8i)3-s + (−55.5 + 96.1i)4-s + (228. + 131. i)5-s + 265. i·6-s + (124. − 215. i)7-s + 985. i·8-s + (−986. − 1.70e3i)9-s + 1.08e3·10-s − 5.00e3·11-s + (−3.57e3 − 6.20e3i)12-s + (−3.94e3 − 2.27e3i)13-s − 1.02e3i·14-s + (−1.47e4 + 8.49e3i)15-s + (−5.07e3 − 8.78e3i)16-s + (3.72e3 − 2.15e3i)17-s + ⋯ |
| L(s) = 1 | + (0.315 − 0.182i)2-s + (−0.689 + 1.19i)3-s + (−0.433 + 0.751i)4-s + (0.816 + 0.471i)5-s + 0.502i·6-s + (0.137 − 0.237i)7-s + 0.680i·8-s + (−0.451 − 0.781i)9-s + 0.343·10-s − 1.13·11-s + (−0.598 − 1.03i)12-s + (−0.497 − 0.287i)13-s − 0.100i·14-s + (−1.12 + 0.650i)15-s + (−0.309 − 0.536i)16-s + (0.183 − 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.111i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0556842 - 0.996874i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0556842 - 0.996874i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-1.03e5 - 2.90e5i)T \) |
| good | 2 | \( 1 + (-3.57 + 2.06i)T + (64 - 110. i)T^{2} \) |
| 3 | \( 1 + (32.2 - 55.8i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-228. - 131. i)T + (3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-124. + 215. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + 5.00e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (3.94e3 + 2.27e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-3.72e3 + 2.15e3i)T + (2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-2.03e4 - 1.17e4i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 - 1.77e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 7.57e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 4.74e4iT - 2.75e10T^{2} \) |
| 41 | \( 1 + (1.49e5 - 2.58e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 - 6.09e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 3.10e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-1.82e5 - 3.16e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-2.52e5 + 1.45e5i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (3.05e6 + 1.76e6i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.85e6 - 3.21e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.96e6 - 5.13e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 2.26e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-1.77e6 - 1.02e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.13e6 + 1.95e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (3.75e6 - 2.16e6i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 4.97e6iT - 8.07e13T^{2} \) |
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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
−15.57129798800220108911734866462, −14.19880448763438541294084320549, −13.13189909335305088665682346814, −11.69900650763677410250502543916, −10.46447173048132979858353457717, −9.638034948190742183169714259718, −7.79483372456117360535094753578, −5.64031804787750697765045698425, −4.56429674035863689015136522288, −2.91131351107933161191962078555,
0.42913663010130905302325076191, 1.86831467356848237537347015747, 5.12099988670478399436387214307, 5.89440745456435263907020700107, 7.32090887062870421198924466202, 9.178314654364727971699422324922, 10.53673574181064936592304734010, 12.16533974002380512165054296261, 13.15668948874009544945126033625, 13.84503630024035163405750572370
