| L(s) = 1 | + (−0.119 + 1.72i)3-s + (0.307 + 0.111i)5-s + (−0.551 + 3.13i)7-s + (−2.97 − 0.412i)9-s + (−0.395 + 0.144i)11-s + (−0.214 + 0.179i)13-s + (−0.230 + 0.518i)15-s + (0.609 + 1.05i)17-s + (−3.01 + 5.21i)19-s + (−5.34 − 1.32i)21-s + (−0.405 − 2.29i)23-s + (−3.74 − 3.14i)25-s + (1.06 − 5.08i)27-s + (5.45 + 4.57i)29-s + (−0.151 − 0.860i)31-s + ⋯ |
| L(s) = 1 | + (−0.0689 + 0.997i)3-s + (0.137 + 0.0500i)5-s + (−0.208 + 1.18i)7-s + (−0.990 − 0.137i)9-s + (−0.119 + 0.0434i)11-s + (−0.0594 + 0.0499i)13-s + (−0.0594 + 0.133i)15-s + (0.147 + 0.256i)17-s + (−0.690 + 1.19i)19-s + (−1.16 − 0.289i)21-s + (−0.0844 − 0.479i)23-s + (−0.749 − 0.629i)25-s + (0.205 − 0.978i)27-s + (1.01 + 0.850i)29-s + (−0.0272 − 0.154i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.413682 + 0.993752i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.413682 + 0.993752i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.119 - 1.72i)T \) |
| good | 5 | \( 1 + (-0.307 - 0.111i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.551 - 3.13i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (0.395 - 0.144i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.214 - 0.179i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.609 - 1.05i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.01 - 5.21i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.405 + 2.29i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.45 - 4.57i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.151 + 0.860i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.77 - 8.26i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.02 - 4.21i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (10.4 - 3.80i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.23 + 12.6i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + (-11.0 - 4.01i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.688 - 3.90i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.38 + 4.51i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.96 - 5.13i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.237 - 0.411i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.82 - 8.24i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (4.80 + 4.03i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-9.28 + 16.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.53 + 0.922i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.61350299096777077990571906516, −10.15991339469445618874549130772, −10.05093860208987132682707689648, −8.651456765883512255523687771786, −8.323446548996044995260947802510, −6.51681397296752265284034256831, −5.73650212541467499529627123495, −4.76934549771422732166590110904, −3.54430633939046950274553374263, −2.33276160712329492216637027448,
0.67722547095463339993242932036, 2.30367789042140078634523045963, 3.74469774798332086684517399452, 5.11313181508068242523587633037, 6.31957522152057754136968321635, 7.12229397911488669657688740099, 7.83483515483101961068241813715, 8.903127545260173215029542827990, 9.988682890395904915264570925149, 10.94045329743582679885270610393
