| L(s) = 1 | + (1.64 + 0.839i)2-s + (−1.17 − 2.29i)3-s + (0.836 + 1.15i)4-s + (0.644 − 2.14i)5-s − 4.76i·6-s + (−0.287 + 1.81i)7-s + (−0.167 − 1.05i)8-s + (−2.14 + 2.94i)9-s + (2.86 − 2.98i)10-s + (0.162 + 0.223i)11-s + (1.66 − 3.26i)12-s + (2.88 + 5.66i)13-s + (−1.99 + 2.74i)14-s + (−5.67 + 1.02i)15-s + (1.48 − 4.58i)16-s + (0.776 + 4.90i)17-s + ⋯ |
| L(s) = 1 | + (1.16 + 0.593i)2-s + (−0.675 − 1.32i)3-s + (0.418 + 0.575i)4-s + (0.288 − 0.957i)5-s − 1.94i·6-s + (−0.108 + 0.685i)7-s + (−0.0590 − 0.372i)8-s + (−0.713 + 0.982i)9-s + (0.904 − 0.944i)10-s + (0.0490 + 0.0674i)11-s + (0.480 − 0.943i)12-s + (0.800 + 1.57i)13-s + (−0.533 + 0.734i)14-s + (−1.46 + 0.264i)15-s + (0.372 − 1.14i)16-s + (0.188 + 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.53788 - 0.557747i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53788 - 0.557747i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.644 + 2.14i)T \) |
| 31 | \( 1 + (4.16 - 3.69i)T \) |
| good | 2 | \( 1 + (-1.64 - 0.839i)T + (1.17 + 1.61i)T^{2} \) |
| 3 | \( 1 + (1.17 + 2.29i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (0.287 - 1.81i)T + (-6.65 - 2.16i)T^{2} \) |
| 11 | \( 1 + (-0.162 - 0.223i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.88 - 5.66i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.776 - 4.90i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-4.79 + 1.55i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (4.90 - 0.776i)T + (21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (1.62 + 5.00i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-2.61 - 2.61i)T + 37iT^{2} \) |
| 41 | \( 1 + (-0.477 - 1.47i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (2.07 + 1.05i)T + (25.2 + 34.7i)T^{2} \) |
| 47 | \( 1 + (3.68 + 7.22i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.665 + 0.105i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-3.12 - 1.01i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + 1.54iT - 61T^{2} \) |
| 67 | \( 1 + (4.81 - 4.81i)T - 67iT^{2} \) |
| 71 | \( 1 + (-4.04 - 2.93i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.49 - 15.7i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (12.1 + 8.82i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (7.49 + 3.82i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-10.6 + 7.77i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (0.303 - 1.91i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−13.05799312732081501107053123535, −12.06559856039893176286138537085, −11.67426151835166954231955467195, −9.639875808237140124302187954755, −8.420858658214269784527224656273, −7.07101704665472832334980444041, −6.08600650800881321356643811898, −5.55450727444617532544713492987, −4.12423303261226917905075875601, −1.63402389551967434255335077288,
3.12446175908974741048056337593, 3.86465556864496315995042009072, 5.23691462447910024156846564046, 5.94430050605260785800428772725, 7.66682265161900759767765817060, 9.566155643316063535850871174918, 10.49654950691664205126589535094, 10.98998086152510163598763626184, 11.86023729055294836340055622999, 13.17040506212559225476112004088
