L(s) = 1 | + (0.5 + 0.866i)2-s + (1 − 1.73i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 1.99·6-s + (−2 + 1.73i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (−1.5 + 2.59i)11-s + (0.999 + 1.73i)12-s − 13-s + (−2.5 − 0.866i)14-s − 1.99·15-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.577 − 0.999i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + 0.816·6-s + (−0.755 + 0.654i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.452 + 0.783i)11-s + (0.288 + 0.499i)12-s − 0.277·13-s + (−0.668 − 0.231i)14-s − 0.516·15-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11281 + 0.0706192i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11281 + 0.0706192i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 3 | \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−14.57781183609616189848107520147, −13.68579527590435481843316327750, −12.50121732159152632150920036533, −12.24788227275347666382833646586, −9.948938616610033244479183380716, −8.617441261340900804509559804041, −7.60422602125420421213833691831, −6.55221263625625154346273123197, −4.92336720086814895226941027035, −2.73046000169629407206692505395,
3.20395034731392218686573802636, 4.06859797742588904077155461716, 5.95989562751509034618273960297, 7.84833271908281547941958143290, 9.433724438513604545475250100804, 10.19095008837039158422829506066, 11.12291794042038496292391479291, 12.59224917444173072857038973396, 13.71089637602596785375825156220, 14.61777141842794189768454698025