| L(s) = 1 | + (0.246 + 1.39i)2-s + (−1.19 − 1.25i)3-s + (−1.87 + 0.687i)4-s + (0.359 + 1.80i)5-s + (1.45 − 1.97i)6-s + (−3.05 + 1.26i)7-s + (−1.42 − 2.44i)8-s + (−0.144 + 2.99i)9-s + (−2.42 + 0.946i)10-s + (−2.05 + 3.07i)11-s + (3.10 + 1.53i)12-s + (0.534 + 0.106i)13-s + (−2.51 − 3.93i)14-s + (1.83 − 2.60i)15-s + (3.05 − 2.58i)16-s + (−0.827 + 0.827i)17-s + ⋯ |
| L(s) = 1 | + (0.174 + 0.984i)2-s + (−0.689 − 0.723i)3-s + (−0.939 + 0.343i)4-s + (0.160 + 0.807i)5-s + (0.592 − 0.805i)6-s + (−1.15 + 0.477i)7-s + (−0.502 − 0.864i)8-s + (−0.0481 + 0.998i)9-s + (−0.767 + 0.299i)10-s + (−0.620 + 0.928i)11-s + (0.896 + 0.442i)12-s + (0.148 + 0.0295i)13-s + (−0.671 − 1.05i)14-s + (0.474 − 0.673i)15-s + (0.763 − 0.645i)16-s + (−0.200 + 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0708130 + 0.567744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0708130 + 0.567744i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.246 - 1.39i)T \) |
| 3 | \( 1 + (1.19 + 1.25i)T \) |
| good | 5 | \( 1 + (-0.359 - 1.80i)T + (-4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (3.05 - 1.26i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (2.05 - 3.07i)T + (-4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-0.534 - 0.106i)T + (12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (0.827 - 0.827i)T - 17iT^{2} \) |
| 19 | \( 1 + (6.44 + 1.28i)T + (17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (-7.47 - 3.09i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (0.420 - 0.280i)T + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 - 2.79T + 31T^{2} \) |
| 37 | \( 1 + (0.213 + 1.07i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (3.82 - 9.23i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-3.05 + 4.56i)T + (-16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (-7.47 - 7.47i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.02 + 1.35i)T + (20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (10.5 - 2.10i)T + (54.5 - 22.5i)T^{2} \) |
| 61 | \( 1 + (-0.153 + 0.102i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (-1.39 - 2.08i)T + (-25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (5.26 + 12.7i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (3.57 - 8.64i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-9.99 - 9.99i)T + 79iT^{2} \) |
| 83 | \( 1 + (1.76 - 8.89i)T + (-76.6 - 31.7i)T^{2} \) |
| 89 | \( 1 + (4.17 + 10.0i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 15.1iT - 97T^{2} \) |
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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
−12.83425806279559722015445051595, −12.58218198772495124011949307358, −11.00944882476726454656972505863, −10.00736895617616560014035749312, −8.814541393100970903920681301879, −7.45508694668020007664068419549, −6.68277150019395557600196229149, −6.04136107315654423631092671127, −4.73712513841728877312813882894, −2.80378124780662789824424677439,
0.51333749197481787668830085147, 3.12261528573455493900805664531, 4.31920557902639108997079680749, 5.37733404662586984023128685899, 6.47481336168153406974257881339, 8.641734382313066547132296579202, 9.273230092696442127363715366840, 10.49215023651010280322369288445, 10.80269916133724703428325176955, 12.15444501075018853718957555141
