| L(s) = 1 | + (−0.489 + 1.50i)2-s + (15.5 + 0.0191i)3-s + (23.8 + 17.3i)4-s + (−16.6 + 5.40i)5-s + (−7.66 + 23.4i)6-s + (36.5 − 50.2i)7-s + (−78.8 + 57.3i)8-s + (242. + 0.598i)9-s − 27.7i·10-s + (56.2 + 397. i)11-s + (371. + 270. i)12-s + (283. + 92.1i)13-s + (57.9 + 79.7i)14-s + (−259. + 83.9i)15-s + (243. + 750. i)16-s + (−320. − 985. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0866 + 0.266i)2-s + (0.999 + 0.00123i)3-s + (0.745 + 0.541i)4-s + (−0.297 + 0.0967i)5-s + (−0.0869 + 0.266i)6-s + (0.281 − 0.387i)7-s + (−0.435 + 0.316i)8-s + (0.999 + 0.00246i)9-s − 0.0877i·10-s + (0.140 + 0.990i)11-s + (0.744 + 0.542i)12-s + (0.465 + 0.151i)13-s + (0.0790 + 0.108i)14-s + (−0.297 + 0.0963i)15-s + (0.238 + 0.732i)16-s + (−0.268 − 0.826i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.08208 + 0.814495i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08208 + 0.814495i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-15.5 - 0.0191i)T \) |
| 11 | \( 1 + (-56.2 - 397. i)T \) |
| good | 2 | \( 1 + (0.489 - 1.50i)T + (-25.8 - 18.8i)T^{2} \) |
| 5 | \( 1 + (16.6 - 5.40i)T + (2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-36.5 + 50.2i)T + (-5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-283. - 92.1i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (320. + 985. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (821. + 1.13e3i)T + (-7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + 3.74e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (5.35e3 + 3.88e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (153. - 473. i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-219. - 159. i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-3.71e3 + 2.69e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.41e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.15e3 - 2.96e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.25e4 - 4.07e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-5.22e3 + 7.19e3i)T + (-2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (2.29e4 - 7.46e3i)T + (6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + 5.39e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (9.42e3 - 3.06e3i)T + (1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-3.52e3 + 4.85e3i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-9.39e4 - 3.05e4i)T + (2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-3.57e4 - 1.09e5i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 - 1.13e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-5.21e3 + 1.60e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−15.60490038821555525941579756537, −14.90058697661824797520463899951, −13.51422139526853041710861928387, −12.22937607259967623251610737654, −10.82666693175058567839136361363, −9.135820918646697187442861348886, −7.79307942950966918112328621825, −6.87794928474803836555944426203, −4.11137718623348216073468893766, −2.31030943900096756436553060250,
1.70168378988997049458107773359, 3.52096524830232460608359409113, 5.99643419461931580993043732793, 7.79091188385451032453404192387, 9.044483128247583659442611043222, 10.50845932876934599499600641559, 11.70845535997115909153644143684, 13.19114309537160404193211876510, 14.54768618215133454454555281789, 15.37990184561792851274585549787
