L(s) = 1 | + (0.5 + 1.65i)3-s + (2.18 + 1.26i)5-s + 3.37·7-s + (−2.5 + 1.65i)9-s − 3.46i·11-s + (2.87 − 1.65i)13-s + (−1 + 4.25i)15-s + (2.18 + 1.26i)17-s + (−4 − 1.73i)19-s + (1.68 + 5.59i)21-s + (−7.93 + 4.57i)23-s + (0.686 + 1.18i)25-s + (−4 − 3.31i)27-s + (−0.186 − 0.322i)29-s − 7.72i·31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.957i)3-s + (0.977 + 0.564i)5-s + 1.27·7-s + (−0.833 + 0.552i)9-s − 1.04i·11-s + (0.796 − 0.459i)13-s + (−0.258 + 1.09i)15-s + (0.530 + 0.306i)17-s + (−0.917 − 0.397i)19-s + (0.367 + 1.22i)21-s + (−1.65 + 0.954i)23-s + (0.137 + 0.237i)25-s + (−0.769 − 0.638i)27-s + (−0.0345 − 0.0598i)29-s − 1.38i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.70242 + 0.962681i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.70242 + 0.962681i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 1.65i)T \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
good | 5 | \( 1 + (-2.18 - 1.26i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + (-2.87 + 1.65i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.18 - 1.26i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (7.93 - 4.57i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.186 + 0.322i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7.72iT - 31T^{2} \) |
| 37 | \( 1 - 11.1iT - 37T^{2} \) |
| 41 | \( 1 + (-3.18 + 5.51i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.87 - 10.1i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.813 + 0.469i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.81 + 4.87i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.813 + 1.40i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.24 - 0.718i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.18 + 10.7i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.87 - 4.97i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (12.9 + 7.49i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 + (0.813 + 1.40i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.44 - 1.40i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.09819229070369825135912879175, −10.30928556488124770640136180332, −9.615019135761500560466888150202, −8.363620900104168618366679881171, −8.034398319668410910601097867767, −6.15401395277394206028455483486, −5.64132553557512324061305354201, −4.39677866332036171717233014604, −3.24212322480701552686971321537, −1.91554779847672157340912749246,
1.53644162705896159487650076341, 2.14561505086780020413147479851, 4.15204743744142905566071107489, 5.33971725089627094951072839660, 6.23430903613080236487127215312, 7.30899414083107809990268090093, 8.286917593054117441247231444666, 8.874078784444353675045650196185, 9.969993649077656702356229393321, 10.97682311437756779410267149401