| 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
−14.27101188876520863417481732407, −13.26790297750004540764531460726, −12.33703550494799057297995966693, −11.64412321963297774902792225940, −9.569095643084843110345260956933, −8.855337129170740891097230303477, −7.75993390623427248400135232415, −5.35080421307036370973486692685, −4.32192579132361819043541267015, −3.27648327818016003597296667282,
3.23397906905389523645366337410, 4.01965956832450775331631314446, 6.51082632342182893818877791724, 6.97389491786256264879946932498, 8.534525561832457367504963982734, 9.926012670393612661404770448201, 11.42855095444200463667243807473, 12.51091200423231914546085317187, 13.49782905286295183465618487753, 14.57637974296296404121611249621
