L(s) = 1 | + (−0.784 + 0.619i)2-s + (0.210 + 0.977i)3-s + (0.231 − 0.972i)4-s + (−0.811 + 0.584i)5-s + (−0.770 − 0.637i)6-s + (0.662 + 0.749i)7-s + (0.420 + 0.907i)8-s + (−0.911 + 0.410i)9-s + (0.274 − 0.961i)10-s + (0.726 − 0.687i)11-s + (0.999 + 0.0222i)12-s + (0.937 − 0.348i)13-s + (−0.984 − 0.177i)14-s + (−0.741 − 0.670i)15-s + (−0.892 − 0.451i)16-s + (0.338 − 0.940i)17-s + ⋯ |
L(s) = 1 | + (−0.784 + 0.619i)2-s + (0.210 + 0.977i)3-s + (0.231 − 0.972i)4-s + (−0.811 + 0.584i)5-s + (−0.770 − 0.637i)6-s + (0.662 + 0.749i)7-s + (0.420 + 0.907i)8-s + (−0.911 + 0.410i)9-s + (0.274 − 0.961i)10-s + (0.726 − 0.687i)11-s + (0.999 + 0.0222i)12-s + (0.937 − 0.348i)13-s + (−0.984 − 0.177i)14-s + (−0.741 − 0.670i)15-s + (−0.892 − 0.451i)16-s + (0.338 − 0.940i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.068026803 + 0.2205009567i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.068026803 + 0.2205009567i\) |
\(L(1)\) |
\(\approx\) |
\(0.6920483102 + 0.3475994991i\) |
\(L(1)\) |
\(\approx\) |
\(0.6920483102 + 0.3475994991i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 283 | \( 1 \) |
good | 2 | \( 1 + (-0.784 + 0.619i)T \) |
| 3 | \( 1 + (0.210 + 0.977i)T \) |
| 5 | \( 1 + (-0.811 + 0.584i)T \) |
| 7 | \( 1 + (0.662 + 0.749i)T \) |
| 11 | \( 1 + (0.726 - 0.687i)T \) |
| 13 | \( 1 + (0.937 - 0.348i)T \) |
| 17 | \( 1 + (0.338 - 0.940i)T \) |
| 19 | \( 1 + (0.166 - 0.986i)T \) |
| 23 | \( 1 + (-0.679 - 0.734i)T \) |
| 29 | \( 1 + (-0.538 + 0.842i)T \) |
| 31 | \( 1 + (0.610 - 0.791i)T \) |
| 37 | \( 1 + (0.0779 - 0.996i)T \) |
| 41 | \( 1 + (-0.575 - 0.818i)T \) |
| 43 | \( 1 + (-0.359 - 0.933i)T \) |
| 47 | \( 1 + (0.770 - 0.637i)T \) |
| 53 | \( 1 + (0.892 - 0.451i)T \) |
| 59 | \( 1 + (-0.798 - 0.602i)T \) |
| 61 | \( 1 + (0.695 - 0.718i)T \) |
| 67 | \( 1 + (-0.480 + 0.876i)T \) |
| 71 | \( 1 + (0.231 + 0.972i)T \) |
| 73 | \( 1 + (-0.610 - 0.791i)T \) |
| 79 | \( 1 + (-0.359 + 0.933i)T \) |
| 83 | \( 1 + (-0.993 + 0.111i)T \) |
| 89 | \( 1 + (0.628 - 0.777i)T \) |
| 97 | \( 1 + (0.975 - 0.220i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.36497965983921952770455911374, −24.47866588411910480296609118999, −23.53993797262050454145147190642, −22.83981211146707235179711525853, −21.21525973383590931503467340625, −20.40756789934579339814943296849, −19.81509147855521695516744989545, −19.03920793519630417474630708202, −18.09668873274008454893077726733, −17.18513924223475736527580801181, −16.56277558778691307702343613507, −15.14880099383894219180658666520, −13.89284376175289803065116058513, −12.9434009047088303986439597844, −11.91887025105750274828051451235, −11.52006932616992247613764039842, −10.20080410219277259083096504171, −8.892399914157431829801573155036, −8.04459709917987909808023776809, −7.52249769343320787673385111303, −6.29059203482071505787252165292, −4.284256789453002028003174498524, −3.48436227800026652008284971441, −1.58344218481446269744864998758, −1.20616174370467281217137553427,
0.491452263518627522742045543249, 2.46405702305963152214551345147, 3.76503513072120933178405099849, 5.07110607559186774320949190953, 6.062346610746620438759676538966, 7.36666618908744042435390452683, 8.53744028139273108206664254468, 8.90834071497052361973939145474, 10.25519066285726610917548831085, 11.20722937026468832204432036442, 11.6777373935969009401301400705, 13.94221176497461879856373560442, 14.587105668683014264805971079880, 15.53874022730849018555499149723, 15.95459121949704086647124962884, 17.0091986793269691503181507533, 18.20987591447314629776734013064, 18.8371281774314123683309845965, 19.93731430443721588788315158854, 20.63887420396793206890867845396, 21.94595140258706994448994391772, 22.66606615201450348063738005560, 23.75652984405730925062972899922, 24.70234114824610264135066547712, 25.603869504476180578771609271339