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
−16.01192918512774543447531145720, −13.62675418639613981599789918933, −13.11716653272814815805826240735, −12.40300611035162886958265170779, −10.35007232142493528803835602108, −9.051786089124559239025580056952, −7.63456580941150225697844913384, −5.57607910263149803177367457078, −4.11312595697923232349108658563, −0.64878885594183484794609197470,
3.14544663710278357829607012254, 5.60336650725672745882295430012, 6.41904838605115387929609729481, 8.970686673188388795159198534835, 10.13404053037174349840310472877, 11.12122715009901339077192137154, 12.90585518640613575324981138321, 14.07325602190335815211933645616, 15.15992014415900543060551773275, 16.06703256269667540654224236583