| L(s) = 1 | + (0.619 + 0.201i)2-s + (−4.20 + 3.05i)3-s + (−12.6 − 9.15i)4-s + (12.5 + 38.5i)5-s + (−3.22 + 1.04i)6-s + (−44.8 + 61.7i)7-s + (−12.0 − 16.6i)8-s + (8.34 − 25.6i)9-s + 26.4i·10-s + (−96.5 − 72.9i)11-s + 80.9·12-s + (181. + 58.9i)13-s + (−40.2 + 29.2i)14-s + (−170. − 123. i)15-s + (72.8 + 224. i)16-s + (116. − 37.8i)17-s + ⋯ |
| L(s) = 1 | + (0.154 + 0.0503i)2-s + (−0.467 + 0.339i)3-s + (−0.787 − 0.572i)4-s + (0.501 + 1.54i)5-s + (−0.0894 + 0.0290i)6-s + (−0.914 + 1.25i)7-s + (−0.189 − 0.260i)8-s + (0.103 − 0.317i)9-s + 0.264i·10-s + (−0.797 − 0.602i)11-s + 0.562·12-s + (1.07 + 0.349i)13-s + (−0.205 + 0.149i)14-s + (−0.757 − 0.550i)15-s + (0.284 + 0.875i)16-s + (0.403 − 0.130i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.404025 + 0.754744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.404025 + 0.754744i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.20 - 3.05i)T \) |
| 11 | \( 1 + (96.5 + 72.9i)T \) |
| good | 2 | \( 1 + (-0.619 - 0.201i)T + (12.9 + 9.40i)T^{2} \) |
| 5 | \( 1 + (-12.5 - 38.5i)T + (-505. + 367. i)T^{2} \) |
| 7 | \( 1 + (44.8 - 61.7i)T + (-741. - 2.28e3i)T^{2} \) |
| 13 | \( 1 + (-181. - 58.9i)T + (2.31e4 + 1.67e4i)T^{2} \) |
| 17 | \( 1 + (-116. + 37.8i)T + (6.75e4 - 4.90e4i)T^{2} \) |
| 19 | \( 1 + (45.9 + 63.2i)T + (-4.02e4 + 1.23e5i)T^{2} \) |
| 23 | \( 1 - 262.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (13.2 - 18.1i)T + (-2.18e5 - 6.72e5i)T^{2} \) |
| 31 | \( 1 + (385. - 1.18e3i)T + (-7.47e5 - 5.42e5i)T^{2} \) |
| 37 | \( 1 + (361. + 262. i)T + (5.79e5 + 1.78e6i)T^{2} \) |
| 41 | \( 1 + (-1.07e3 - 1.48e3i)T + (-8.73e5 + 2.68e6i)T^{2} \) |
| 43 | \( 1 - 3.04e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-3.21e3 + 2.33e3i)T + (1.50e6 - 4.64e6i)T^{2} \) |
| 53 | \( 1 + (116. - 359. i)T + (-6.38e6 - 4.63e6i)T^{2} \) |
| 59 | \( 1 + (2.43e3 + 1.76e3i)T + (3.74e6 + 1.15e7i)T^{2} \) |
| 61 | \( 1 + (-3.61e3 + 1.17e3i)T + (1.12e7 - 8.13e6i)T^{2} \) |
| 67 | \( 1 + 618.T + 2.01e7T^{2} \) |
| 71 | \( 1 + (1.32e3 + 4.06e3i)T + (-2.05e7 + 1.49e7i)T^{2} \) |
| 73 | \( 1 + (4.75e3 - 6.55e3i)T + (-8.77e6 - 2.70e7i)T^{2} \) |
| 79 | \( 1 + (-665. - 216. i)T + (3.15e7 + 2.28e7i)T^{2} \) |
| 83 | \( 1 + (157. - 51.3i)T + (3.83e7 - 2.78e7i)T^{2} \) |
| 89 | \( 1 - 3.55e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-3.52e3 + 1.08e4i)T + (-7.16e7 - 5.20e7i)T^{2} \) |
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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
−16.06703256269667540654224236583, −15.15992014415900543060551773275, −14.07325602190335815211933645616, −12.90585518640613575324981138321, −11.12122715009901339077192137154, −10.13404053037174349840310472877, −8.970686673188388795159198534835, −6.41904838605115387929609729481, −5.60336650725672745882295430012, −3.14544663710278357829607012254,
0.64878885594183484794609197470, 4.11312595697923232349108658563, 5.57607910263149803177367457078, 7.63456580941150225697844913384, 9.051786089124559239025580056952, 10.35007232142493528803835602108, 12.40300611035162886958265170779, 13.11716653272814815805826240735, 13.62675418639613981599789918933, 16.01192918512774543447531145720
