| L(s) = 1 | + (0.962 + 1.44i)3-s + (2.75 − 2.31i)5-s + (1.28 − 0.467i)7-s + (−1.14 + 2.77i)9-s + (0.884 + 0.742i)11-s + (−1.09 − 6.20i)13-s + (5.97 + 1.74i)15-s + (0.526 + 0.911i)17-s + (−1.05 + 1.82i)19-s + (1.91 + 1.40i)21-s + (−6.16 − 2.24i)23-s + (1.37 − 7.80i)25-s + (−5.09 + 1.01i)27-s + (−1.37 + 7.80i)29-s + (5.95 + 2.16i)31-s + ⋯ |
| L(s) = 1 | + (0.555 + 0.831i)3-s + (1.23 − 1.03i)5-s + (0.485 − 0.176i)7-s + (−0.382 + 0.923i)9-s + (0.266 + 0.223i)11-s + (−0.303 − 1.72i)13-s + (1.54 + 0.449i)15-s + (0.127 + 0.221i)17-s + (−0.241 + 0.418i)19-s + (0.416 + 0.305i)21-s + (−1.28 − 0.467i)23-s + (0.275 − 1.56i)25-s + (−0.980 + 0.195i)27-s + (−0.255 + 1.44i)29-s + (1.07 + 0.389i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.02255 + 0.139891i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.02255 + 0.139891i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.962 - 1.44i)T \) |
| good | 5 | \( 1 + (-2.75 + 2.31i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.28 + 0.467i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.884 - 0.742i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.09 + 6.20i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.526 - 0.911i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.05 - 1.82i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.16 + 2.24i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.37 - 7.80i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-5.95 - 2.16i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-4.41 - 7.64i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.448 - 2.54i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.115 + 0.0967i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (9.75 - 3.54i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 2.09T + 53T^{2} \) |
| 59 | \( 1 + (3.16 - 2.65i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-8.44 + 3.07i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.79 + 10.1i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (8.05 + 13.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.63 + 2.83i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.318 - 1.80i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (1.68 - 9.55i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (3.14 - 5.44i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.4 + 8.73i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.80853411063031003377955124277, −10.06029741090691155525996855409, −9.563492995797546694360330137138, −8.402207713065570795397460823992, −7.997549441538947444010233506290, −6.16867920853352002268659858348, −5.21821507960033539197994568496, −4.54180758636471609163356004773, −3.02518221327902061130013177088, −1.60162435812684700065749241118,
1.86775661087753459414075473104, 2.51007363212248952043432992869, 4.08124106845284210754921388287, 5.80405892936496132358995848767, 6.50505059304830907949169668004, 7.27343292087999108736875668909, 8.389264333602290368037504923645, 9.438296207330063914838646369357, 9.961883689418177371812727524515, 11.44497384545422543930079856534
