| L(s) = 1 | + (−0.0718 + 0.213i)2-s + (2.81 − 1.02i)3-s + (3.14 + 2.39i)4-s + (−7.60 − 3.03i)5-s + (0.0157 + 0.675i)6-s + (4.35 − 2.01i)7-s + (−1.48 + 1.00i)8-s + (6.90 − 5.77i)9-s + (1.19 − 1.40i)10-s + (20.3 + 5.64i)11-s + (11.3 + 3.52i)12-s + (10.6 − 20.1i)13-s + (0.116 + 1.07i)14-s + (−24.5 − 0.757i)15-s + (4.11 + 14.8i)16-s + (−6.72 + 14.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.0359 + 0.106i)2-s + (0.939 − 0.341i)3-s + (0.786 + 0.597i)4-s + (−1.52 − 0.606i)5-s + (0.00262 + 0.112i)6-s + (0.622 − 0.287i)7-s + (−0.185 + 0.125i)8-s + (0.766 − 0.641i)9-s + (0.119 − 0.140i)10-s + (1.84 + 0.512i)11-s + (0.942 + 0.293i)12-s + (0.820 − 1.54i)13-s + (0.00834 + 0.0767i)14-s + (−1.63 − 0.0504i)15-s + (0.257 + 0.927i)16-s + (−0.395 + 0.855i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.06076 - 0.265935i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.06076 - 0.265935i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.81 + 1.02i)T \) |
| 59 | \( 1 + (58.4 - 8.25i)T \) |
| good | 2 | \( 1 + (0.0718 - 0.213i)T + (-3.18 - 2.42i)T^{2} \) |
| 5 | \( 1 + (7.60 + 3.03i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (-4.35 + 2.01i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (-20.3 - 5.64i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (-10.6 + 20.1i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (6.72 - 14.5i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (20.7 - 4.57i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (3.12 + 0.511i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (-0.204 - 0.605i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (39.7 + 8.75i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (15.8 - 23.3i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (13.9 - 2.29i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (3.08 + 11.1i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (11.3 - 4.53i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (-35.8 + 30.4i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (-14.2 - 4.80i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-29.3 - 43.3i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (32.2 - 12.8i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (58.7 - 6.39i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (-40.6 + 24.4i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (-15.8 + 0.860i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (-6.15 - 18.2i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (-147. - 15.9i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−12.42915826211329963261018045872, −11.65242737849568606810309027584, −10.66504640617608279376913385049, −8.800299797313097945767819053981, −8.262552748181623435485623601707, −7.52997170882515810769462504332, −6.48159253530960973029383488919, −4.12756014934150158236823844215, −3.57537000693197100635040361156, −1.51441996102536242744912086907,
1.82450283835384455609393205685, 3.50387405636297909921727204565, 4.38201598162933141935708087481, 6.54727734940539987994451402047, 7.23041311175247816991955481046, 8.595429121017505524544746606189, 9.240675167409390855157818425068, 10.90475261525393884514580365692, 11.34537816799851559692903014906, 12.05701035708902700854694953883
