| L(s) = 1 | + (−0.968 + 1.03i)2-s + (−0.980 + 0.195i)3-s + (−0.122 − 1.99i)4-s + (−0.0578 − 0.0866i)5-s + (0.749 − 1.19i)6-s + (2.40 + 0.997i)7-s + (2.17 + 1.80i)8-s + (0.923 − 0.382i)9-s + (0.145 + 0.0243i)10-s + (−1.19 + 5.98i)11-s + (0.509 + 1.93i)12-s + (−0.590 + 0.883i)13-s + (−3.36 + 1.51i)14-s + (0.0736 + 0.0736i)15-s + (−3.97 + 0.487i)16-s + (0.156 − 0.156i)17-s + ⋯ |
| L(s) = 1 | + (−0.685 + 0.728i)2-s + (−0.566 + 0.112i)3-s + (−0.0610 − 0.998i)4-s + (−0.0258 − 0.0387i)5-s + (0.305 − 0.489i)6-s + (0.910 + 0.377i)7-s + (0.768 + 0.639i)8-s + (0.307 − 0.127i)9-s + (0.0459 + 0.00768i)10-s + (−0.359 + 1.80i)11-s + (0.146 + 0.558i)12-s + (−0.163 + 0.245i)13-s + (−0.898 + 0.404i)14-s + (0.0190 + 0.0190i)15-s + (−0.992 + 0.121i)16-s + (0.0379 − 0.0379i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0849 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0849 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.491824 + 0.535556i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.491824 + 0.535556i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.968 - 1.03i)T \) |
| 3 | \( 1 + (0.980 - 0.195i)T \) |
| good | 5 | \( 1 + (0.0578 + 0.0866i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (-2.40 - 0.997i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (1.19 - 5.98i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (0.590 - 0.883i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-0.156 + 0.156i)T - 17iT^{2} \) |
| 19 | \( 1 + (-3.11 - 2.08i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-0.522 - 1.26i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-1.10 - 5.54i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 7.51iT - 31T^{2} \) |
| 37 | \( 1 + (-6.15 + 4.11i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (1.10 + 2.66i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (6.90 + 1.37i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-8.09 + 8.09i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.714 + 3.59i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (6.55 + 9.80i)T + (-22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (2.79 - 0.555i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-0.861 + 0.171i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (7.68 + 3.18i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (5.79 - 2.39i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-12.1 - 12.1i)T + 79iT^{2} \) |
| 83 | \( 1 + (7.21 + 4.82i)T + (31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (-3.71 + 8.96i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 4.45iT - 97T^{2} \) |
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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
−12.57938403807233805169894255760, −11.74784025761406094051057159239, −10.57898620660380488688938780482, −9.851799877421145206130637521122, −8.738380940486835369533725815510, −7.60051552368642261558123092056, −6.80488160742756362611101899743, −5.33274991430991760805362716757, −4.67746200359123720679424380151, −1.80172804236876633223542538423,
0.956590151824249191691482786160, 2.96845087224810898852249835108, 4.52996697122977225047302466028, 5.94749460534592897195520764832, 7.51874076457194358556703202962, 8.216025196267714064155966766690, 9.398776602476101066906623256500, 10.63241184023283089010052896627, 11.22949481070676939451846474369, 11.82760697768985286988876470262
