| L(s) = 1 | + (1.32 + 0.314i)2-s + (1.39 − 1.02i)3-s + (−0.121 − 0.0609i)4-s + (−2.27 + 3.05i)5-s + (2.17 − 0.927i)6-s + (−1.45 − 0.959i)7-s + (−2.23 − 1.87i)8-s + (0.884 − 2.86i)9-s + (−3.98 + 3.34i)10-s + (3.03 − 0.355i)11-s + (−0.231 + 0.0398i)12-s + (−0.535 + 1.78i)13-s + (−1.63 − 1.73i)14-s + (−0.0278 + 6.59i)15-s + (−2.21 − 2.97i)16-s + (1.21 + 6.89i)17-s + ⋯ |
| L(s) = 1 | + (0.939 + 0.222i)2-s + (0.804 − 0.593i)3-s + (−0.0607 − 0.0304i)4-s + (−1.01 + 1.36i)5-s + (0.888 − 0.378i)6-s + (−0.551 − 0.362i)7-s + (−0.789 − 0.662i)8-s + (0.294 − 0.955i)9-s + (−1.26 + 1.05i)10-s + (0.916 − 0.107i)11-s + (−0.0669 + 0.0115i)12-s + (−0.148 + 0.496i)13-s + (−0.437 − 0.463i)14-s + (−0.00717 + 1.70i)15-s + (−0.553 − 0.743i)16-s + (0.294 + 1.67i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.41247 + 0.0154885i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41247 + 0.0154885i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.39 + 1.02i)T \) |
| good | 2 | \( 1 + (-1.32 - 0.314i)T + (1.78 + 0.897i)T^{2} \) |
| 5 | \( 1 + (2.27 - 3.05i)T + (-1.43 - 4.78i)T^{2} \) |
| 7 | \( 1 + (1.45 + 0.959i)T + (2.77 + 6.42i)T^{2} \) |
| 11 | \( 1 + (-3.03 + 0.355i)T + (10.7 - 2.53i)T^{2} \) |
| 13 | \( 1 + (0.535 - 1.78i)T + (-10.8 - 7.14i)T^{2} \) |
| 17 | \( 1 + (-1.21 - 6.89i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.858 + 4.86i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (-1.59 + 1.04i)T + (9.10 - 21.1i)T^{2} \) |
| 29 | \( 1 + (0.155 - 0.164i)T + (-1.68 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-0.229 - 3.93i)T + (-30.7 + 3.59i)T^{2} \) |
| 37 | \( 1 + (-0.346 - 0.126i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (4.21 - 0.998i)T + (36.6 - 18.4i)T^{2} \) |
| 43 | \( 1 + (1.28 - 2.98i)T + (-29.5 - 31.2i)T^{2} \) |
| 47 | \( 1 + (-0.101 + 1.74i)T + (-46.6 - 5.45i)T^{2} \) |
| 53 | \( 1 + (3.03 + 5.25i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.98 + 0.231i)T + (57.4 + 13.6i)T^{2} \) |
| 61 | \( 1 + (-11.4 + 5.73i)T + (36.4 - 48.9i)T^{2} \) |
| 67 | \( 1 + (-4.35 - 4.61i)T + (-3.89 + 66.8i)T^{2} \) |
| 71 | \( 1 + (10.6 - 8.97i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.42 - 2.03i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (2.55 + 0.604i)T + (70.5 + 35.4i)T^{2} \) |
| 83 | \( 1 + (-0.218 - 0.0518i)T + (74.1 + 37.2i)T^{2} \) |
| 89 | \( 1 + (5.45 + 4.57i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-9.30 - 12.4i)T + (-27.8 + 92.9i)T^{2} \) |
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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
−14.57637974296296404121611249621, −13.49782905286295183465618487753, −12.51091200423231914546085317187, −11.42855095444200463667243807473, −9.926012670393612661404770448201, −8.534525561832457367504963982734, −6.97389491786256264879946932498, −6.51082632342182893818877791724, −4.01965956832450775331631314446, −3.23397906905389523645366337410,
3.27648327818016003597296667282, 4.32192579132361819043541267015, 5.35080421307036370973486692685, 7.75993390623427248400135232415, 8.855337129170740891097230303477, 9.569095643084843110345260956933, 11.64412321963297774902792225940, 12.33703550494799057297995966693, 13.26790297750004540764531460726, 14.27101188876520863417481732407
