| L(s) = 1 | + (−0.308 − 0.743i)2-s + (3.07 − 2.05i)3-s + (10.8 − 10.8i)4-s + (11.5 + 2.29i)5-s + (−2.47 − 1.65i)6-s + (−14.9 + 2.96i)7-s + (−23.3 − 9.66i)8-s + (−25.7 + 62.2i)9-s + (−1.84 − 9.27i)10-s + (−65.8 + 98.5i)11-s + (11.0 − 55.6i)12-s + (−22.6 − 22.6i)13-s + (6.80 + 10.1i)14-s + (40.0 − 16.6i)15-s − 225. i·16-s + (262. + 121. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0770 − 0.185i)2-s + (0.341 − 0.228i)3-s + (0.678 − 0.678i)4-s + (0.460 + 0.0916i)5-s + (−0.0687 − 0.0459i)6-s + (−0.304 + 0.0605i)7-s + (−0.364 − 0.150i)8-s + (−0.318 + 0.768i)9-s + (−0.0184 − 0.0927i)10-s + (−0.544 + 0.814i)11-s + (0.0768 − 0.386i)12-s + (−0.133 − 0.133i)13-s + (0.0347 + 0.0519i)14-s + (0.178 − 0.0738i)15-s − 0.880i·16-s + (0.906 + 0.421i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.33519 - 0.403845i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33519 - 0.403845i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-262. - 121. i)T \) |
| good | 2 | \( 1 + (0.308 + 0.743i)T + (-11.3 + 11.3i)T^{2} \) |
| 3 | \( 1 + (-3.07 + 2.05i)T + (30.9 - 74.8i)T^{2} \) |
| 5 | \( 1 + (-11.5 - 2.29i)T + (577. + 239. i)T^{2} \) |
| 7 | \( 1 + (14.9 - 2.96i)T + (2.21e3 - 918. i)T^{2} \) |
| 11 | \( 1 + (65.8 - 98.5i)T + (-5.60e3 - 1.35e4i)T^{2} \) |
| 13 | \( 1 + (22.6 + 22.6i)T + 2.85e4iT^{2} \) |
| 19 | \( 1 + (30.7 + 74.1i)T + (-9.21e4 + 9.21e4i)T^{2} \) |
| 23 | \( 1 + (-1.60 - 1.07i)T + (1.07e5 + 2.58e5i)T^{2} \) |
| 29 | \( 1 + (84.4 - 424. i)T + (-6.53e5 - 2.70e5i)T^{2} \) |
| 31 | \( 1 + (601. + 900. i)T + (-3.53e5 + 8.53e5i)T^{2} \) |
| 37 | \( 1 + (-1.67e3 + 1.11e3i)T + (7.17e5 - 1.73e6i)T^{2} \) |
| 41 | \( 1 + (-1.48e3 + 294. i)T + (2.61e6 - 1.08e6i)T^{2} \) |
| 43 | \( 1 + (-1.13e3 + 2.74e3i)T + (-2.41e6 - 2.41e6i)T^{2} \) |
| 47 | \( 1 + (1.44e3 + 1.44e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.39e3 - 3.37e3i)T + (-5.57e6 + 5.57e6i)T^{2} \) |
| 59 | \( 1 + (5.53e3 + 2.29e3i)T + (8.56e6 + 8.56e6i)T^{2} \) |
| 61 | \( 1 + (-1.02e3 - 5.16e3i)T + (-1.27e7 + 5.29e6i)T^{2} \) |
| 67 | \( 1 + 23.3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (-5.73e3 + 3.82e3i)T + (9.72e6 - 2.34e7i)T^{2} \) |
| 73 | \( 1 + (-7.28e3 - 1.44e3i)T + (2.62e7 + 1.08e7i)T^{2} \) |
| 79 | \( 1 + (4.61e3 - 6.91e3i)T + (-1.49e7 - 3.59e7i)T^{2} \) |
| 83 | \( 1 + (1.84e3 - 763. i)T + (3.35e7 - 3.35e7i)T^{2} \) |
| 89 | \( 1 + (5.87e3 - 5.87e3i)T - 6.27e7iT^{2} \) |
| 97 | \( 1 + (976. - 4.90e3i)T + (-8.17e7 - 3.38e7i)T^{2} \) |
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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
−18.28942935939114509736020019117, −16.67362399563207583374917176442, −15.28414572441996449541967323817, −14.10329460597852557342045410377, −12.60902397031605037825617806506, −10.86028184967935191860712491452, −9.679437376757765960554703533125, −7.53822261888031778247770303545, −5.68435135896524245054153391705, −2.27751368656430920491997255089,
3.15789128127361071669024052562, 6.14026143009991549931471744038, 7.993052885809533750206876496167, 9.574072651639399773506906558967, 11.41465350440214573001339965670, 12.83926820899062588985827468307, 14.39415083961466042918842038480, 15.81911503882573407008596622388, 16.81304161633545229198519484827, 18.09029788952499830886595494987
