| L(s) = 1 | + (0.841 − 0.540i)2-s − 3-s + (0.415 − 0.909i)4-s + (−0.149 + 1.04i)5-s + (−0.841 + 0.540i)6-s + (−1.49 + 1.72i)7-s + (−0.142 − 0.989i)8-s + 9-s + (0.437 + 0.957i)10-s + (1.40 − 3.00i)11-s + (−0.415 + 0.909i)12-s + (0.661 − 1.44i)13-s + (−0.325 + 2.26i)14-s + (0.149 − 1.04i)15-s + (−0.654 − 0.755i)16-s + (4.69 + 1.37i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s − 0.577·3-s + (0.207 − 0.454i)4-s + (−0.0670 + 0.466i)5-s + (−0.343 + 0.220i)6-s + (−0.565 + 0.652i)7-s + (−0.0503 − 0.349i)8-s + 0.333·9-s + (0.138 + 0.302i)10-s + (0.422 − 0.906i)11-s + (−0.119 + 0.262i)12-s + (0.183 − 0.401i)13-s + (−0.0869 + 0.604i)14-s + (0.0386 − 0.269i)15-s + (−0.163 − 0.188i)16-s + (1.13 + 0.334i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.75584 - 0.336400i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75584 - 0.336400i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + (-1.40 + 3.00i)T \) |
| good | 5 | \( 1 + (0.149 - 1.04i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (1.49 - 1.72i)T + (-0.996 - 6.92i)T^{2} \) |
| 13 | \( 1 + (-0.661 + 1.44i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-4.69 - 1.37i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-6.04 + 1.77i)T + (15.9 - 10.2i)T^{2} \) |
| 23 | \( 1 + (-4.60 - 5.31i)T + (-3.27 + 22.7i)T^{2} \) |
| 29 | \( 1 + (-0.0196 + 0.00575i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (1.08 + 2.38i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.73 - 5.99i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (6.78 - 4.35i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (1.52 + 10.6i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (-2.09 - 1.34i)T + (19.5 + 42.7i)T^{2} \) |
| 53 | \( 1 + (-8.34 + 9.63i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (10.5 + 6.77i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.91 - 2.51i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-1.64 + 1.05i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (10.6 - 3.11i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (0.0828 + 0.0956i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.19 + 8.29i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (7.82 - 9.03i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (14.6 + 4.29i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (0.0583 + 0.406i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.46731147676428018371873157693, −9.722304683423664497075876888852, −8.811365782137178926856578024457, −7.51737006532400002080182169504, −6.59817355323234714365061617616, −5.70067560402407630003505049948, −5.16360212616332046296405136465, −3.50652612799112537452393743907, −3.03037591559667100122533720048, −1.14612028950410412750503684939,
1.15160750831593481524265721499, 3.09681701573885182612105512939, 4.21942223960234454818245099428, 4.99582129800641690717021475386, 5.94398441090161068276234640206, 7.01917087806644799810067821542, 7.39433783028552366363236867921, 8.736429731018612095715230913792, 9.683082343920985610459731605461, 10.43150347407653299407324254250
