| L(s) = 1 | + (4.09 − 1.33i)2-s + (−8.24 − 5.98i)3-s + (2.04 − 1.48i)4-s + (3.45 − 10.6i)5-s + (−41.7 − 13.5i)6-s + (−31.1 − 42.9i)7-s + (−34.0 + 46.9i)8-s + (7.03 + 21.6i)9-s − 48.1i·10-s + (−53.9 − 108. i)11-s − 25.7·12-s + (−114. + 37.3i)13-s + (−184. − 134. i)14-s + (−92.1 + 66.9i)15-s + (−89.6 + 275. i)16-s + (433. + 140. i)17-s + ⋯ |
| L(s) = 1 | + (1.02 − 0.332i)2-s + (−0.915 − 0.665i)3-s + (0.127 − 0.0928i)4-s + (0.138 − 0.425i)5-s + (−1.15 − 0.376i)6-s + (−0.636 − 0.875i)7-s + (−0.532 + 0.733i)8-s + (0.0868 + 0.267i)9-s − 0.481i·10-s + (−0.445 − 0.895i)11-s − 0.178·12-s + (−0.680 + 0.221i)13-s + (−0.942 − 0.684i)14-s + (−0.409 + 0.297i)15-s + (−0.350 + 1.07i)16-s + (1.50 + 0.487i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.465i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.885 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.296119 - 1.19997i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.296119 - 1.19997i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-3.45 + 10.6i)T \) |
| 11 | \( 1 + (53.9 + 108. i)T \) |
| good | 2 | \( 1 + (-4.09 + 1.33i)T + (12.9 - 9.40i)T^{2} \) |
| 3 | \( 1 + (8.24 + 5.98i)T + (25.0 + 77.0i)T^{2} \) |
| 7 | \( 1 + (31.1 + 42.9i)T + (-741. + 2.28e3i)T^{2} \) |
| 13 | \( 1 + (114. - 37.3i)T + (2.31e4 - 1.67e4i)T^{2} \) |
| 17 | \( 1 + (-433. - 140. i)T + (6.75e4 + 4.90e4i)T^{2} \) |
| 19 | \( 1 + (-379. + 521. i)T + (-4.02e4 - 1.23e5i)T^{2} \) |
| 23 | \( 1 - 516.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (627. + 863. i)T + (-2.18e5 + 6.72e5i)T^{2} \) |
| 31 | \( 1 + (307. + 946. i)T + (-7.47e5 + 5.42e5i)T^{2} \) |
| 37 | \( 1 + (-632. + 459. i)T + (5.79e5 - 1.78e6i)T^{2} \) |
| 41 | \( 1 + (1.57e3 - 2.17e3i)T + (-8.73e5 - 2.68e6i)T^{2} \) |
| 43 | \( 1 - 1.22e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (213. + 155. i)T + (1.50e6 + 4.64e6i)T^{2} \) |
| 53 | \( 1 + (-8.03 - 24.7i)T + (-6.38e6 + 4.63e6i)T^{2} \) |
| 59 | \( 1 + (207. - 150. i)T + (3.74e6 - 1.15e7i)T^{2} \) |
| 61 | \( 1 + (1.15e3 + 375. i)T + (1.12e7 + 8.13e6i)T^{2} \) |
| 67 | \( 1 + 2.18e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-1.98e3 + 6.11e3i)T + (-2.05e7 - 1.49e7i)T^{2} \) |
| 73 | \( 1 + (1.27e3 + 1.75e3i)T + (-8.77e6 + 2.70e7i)T^{2} \) |
| 79 | \( 1 + (-1.03e4 + 3.35e3i)T + (3.15e7 - 2.28e7i)T^{2} \) |
| 83 | \( 1 + (4.08e3 + 1.32e3i)T + (3.83e7 + 2.78e7i)T^{2} \) |
| 89 | \( 1 - 4.15e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-2.78e3 - 8.58e3i)T + (-7.16e7 + 5.20e7i)T^{2} \) |
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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
−13.47829697404336761702874666348, −13.10403884456470362259510189076, −11.98845347032051383178309994966, −11.13950881280736526158708231151, −9.480220636534473548768306682210, −7.60628268961165563839129254836, −6.10252231203574156853312618872, −5.03406526783494460429529154719, −3.28909671410946805157135667887, −0.58553673881706495590752000612,
3.27706535151716426994331491274, 5.17815624296920620148344260634, 5.61140791069800411008229964453, 7.20659895501259944022153377958, 9.566804392448982874366730985902, 10.29067607990160664617914039796, 12.01974798363982348793986758133, 12.58155330978385567113296523095, 14.12333320930477217509156775949, 15.02799163219079254217785232628
