| L(s) = 1 | + (−0.796 + 1.53i)3-s + (1.12 − 3.08i)5-s + (−3.05 − 0.537i)7-s + (−1.73 − 2.45i)9-s + (−1.75 + 0.637i)11-s + (−3.44 + 2.88i)13-s + (3.84 + 4.18i)15-s + (−5.33 + 3.07i)17-s + (−5.61 − 3.24i)19-s + (3.25 − 4.26i)21-s + (−0.0995 − 0.564i)23-s + (−4.42 − 3.71i)25-s + (5.14 − 0.711i)27-s + (5.05 − 6.02i)29-s + (6.55 − 1.15i)31-s + ⋯ |
| L(s) = 1 | + (−0.459 + 0.888i)3-s + (0.502 − 1.37i)5-s + (−1.15 − 0.203i)7-s + (−0.577 − 0.816i)9-s + (−0.527 + 0.192i)11-s + (−0.954 + 0.800i)13-s + (0.994 + 1.08i)15-s + (−1.29 + 0.746i)17-s + (−1.28 − 0.744i)19-s + (0.710 − 0.930i)21-s + (−0.0207 − 0.117i)23-s + (−0.884 − 0.742i)25-s + (0.990 − 0.136i)27-s + (0.938 − 1.11i)29-s + (1.17 − 0.207i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0765620 - 0.246288i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0765620 - 0.246288i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.796 - 1.53i)T \) |
| good | 5 | \( 1 + (-1.12 + 3.08i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (3.05 + 0.537i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.75 - 0.637i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (3.44 - 2.88i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (5.33 - 3.07i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.61 + 3.24i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0995 + 0.564i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.05 + 6.02i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-6.55 + 1.15i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.51 + 4.35i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.01 - 4.78i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.06 + 8.40i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.29 - 7.32i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 4.40iT - 53T^{2} \) |
| 59 | \( 1 + (1.19 + 0.436i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.757 + 4.29i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (3.35 + 3.99i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.77 - 4.80i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.12 - 8.87i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.530 - 0.631i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.90 + 4.95i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.31 + 1.33i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.55 + 3.47i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.56176778799139826749179667053, −9.826714695913490234230723243224, −9.154691264667166566349830602078, −8.445635523891267381108347208163, −6.71503755382601999955798569706, −5.99991452051450808187558260734, −4.68106752973436450426164999581, −4.30888529417268030841026478208, −2.45366767719177596114005614002, −0.15585611303037120059380402719,
2.39659457430313423034196819540, 3.01672267209113261627821257547, 5.04260892872809820259636443580, 6.33805694297981971639565733427, 6.57910463708238791835909243319, 7.55761560805007254924772012314, 8.721280991183647729090833803412, 10.17785537567579284395478702528, 10.41838446091794388408871558965, 11.50582370938739225248057377749
