L(s) = 1 | − 4.46·2-s + (−14.2 − 6.27i)3-s − 12.0·4-s + 59.8i·5-s + (63.6 + 28.0i)6-s − 169. i·7-s + 196.·8-s + (164. + 179. i)9-s − 267. i·10-s + (358. − 181. i)11-s + (172. + 75.8i)12-s + 838. i·13-s + 756. i·14-s + (375. − 854. i)15-s − 491.·16-s + 512.·17-s + ⋯ |
L(s) = 1 | − 0.788·2-s + (−0.915 − 0.402i)3-s − 0.377·4-s + 1.07i·5-s + (0.722 + 0.317i)6-s − 1.30i·7-s + 1.08·8-s + (0.675 + 0.737i)9-s − 0.845i·10-s + (0.892 − 0.451i)11-s + (0.345 + 0.152i)12-s + 1.37i·13-s + 1.03i·14-s + (0.431 − 0.980i)15-s − 0.479·16-s + 0.430·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0537i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.678855 + 0.0182662i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.678855 + 0.0182662i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (14.2 + 6.27i)T \) |
| 11 | \( 1 + (-358. + 181. i)T \) |
good | 2 | \( 1 + 4.46T + 32T^{2} \) |
| 5 | \( 1 - 59.8iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 169. iT - 1.68e4T^{2} \) |
| 13 | \( 1 - 838. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 512.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.87e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.33e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 1.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.95e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.77e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.85e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.01e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.87e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 5.84e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.70e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.73e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 1.66e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 2.58e3iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 3.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.37e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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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
−16.25933457543787520131277333606, −14.18509637121678244681931781306, −13.50049957366615930989114630225, −11.50762428480007579685850199803, −10.71116875417016340317125821470, −9.491503273792906149867112582281, −7.49390588430729872745110001747, −6.62028013041904014303388694463, −4.28620382905283889518747395942, −1.01164343239908765584285203637,
0.908526356593065660483960061786, 4.58697771361391572085302235848, 5.83938311950348311712782872763, 8.218771600433950187415436760788, 9.276254926477801304838995317981, 10.31813711288188873100931219680, 12.10489858463098564858003482078, 12.72032431689462149562618606398, 14.79787409348360203602265298987, 16.11583704366575379814379626320