L(s) = 1 | + (−0.997 − 0.0667i)2-s + (0.984 + 0.177i)3-s + (0.991 + 0.133i)4-s + (−0.770 − 0.637i)5-s + (−0.970 − 0.242i)6-s + (0.756 − 0.654i)7-s + (−0.979 − 0.199i)8-s + (0.937 + 0.348i)9-s + (0.726 + 0.687i)10-s + (−0.380 + 0.924i)11-s + (0.951 + 0.306i)12-s + (0.274 + 0.961i)13-s + (−0.798 + 0.602i)14-s + (−0.645 − 0.763i)15-s + (0.964 + 0.264i)16-s + (−0.122 + 0.992i)17-s + ⋯ |
L(s) = 1 | + (−0.997 − 0.0667i)2-s + (0.984 + 0.177i)3-s + (0.991 + 0.133i)4-s + (−0.770 − 0.637i)5-s + (−0.970 − 0.242i)6-s + (0.756 − 0.654i)7-s + (−0.979 − 0.199i)8-s + (0.937 + 0.348i)9-s + (0.726 + 0.687i)10-s + (−0.380 + 0.924i)11-s + (0.951 + 0.306i)12-s + (0.274 + 0.961i)13-s + (−0.798 + 0.602i)14-s + (−0.645 − 0.763i)15-s + (0.964 + 0.264i)16-s + (−0.122 + 0.992i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.107740096 - 0.07657914420i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.107740096 - 0.07657914420i\) |
\(L(1)\) |
\(\approx\) |
\(0.9608166809 - 0.04636307764i\) |
\(L(1)\) |
\(\approx\) |
\(0.9608166809 - 0.04636307764i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 283 | \( 1 \) |
good | 2 | \( 1 + (-0.997 - 0.0667i)T \) |
| 3 | \( 1 + (0.984 + 0.177i)T \) |
| 5 | \( 1 + (-0.770 - 0.637i)T \) |
| 7 | \( 1 + (0.756 - 0.654i)T \) |
| 11 | \( 1 + (-0.380 + 0.924i)T \) |
| 13 | \( 1 + (0.274 + 0.961i)T \) |
| 17 | \( 1 + (-0.122 + 0.992i)T \) |
| 19 | \( 1 + (0.695 - 0.718i)T \) |
| 23 | \( 1 + (0.519 - 0.854i)T \) |
| 29 | \( 1 + (0.100 - 0.994i)T \) |
| 31 | \( 1 + (0.975 - 0.220i)T \) |
| 37 | \( 1 + (-0.460 - 0.887i)T \) |
| 41 | \( 1 + (0.662 + 0.749i)T \) |
| 43 | \( 1 + (-0.420 - 0.907i)T \) |
| 47 | \( 1 + (-0.970 + 0.242i)T \) |
| 53 | \( 1 + (0.964 - 0.264i)T \) |
| 59 | \( 1 + (-0.929 + 0.369i)T \) |
| 61 | \( 1 + (0.231 + 0.972i)T \) |
| 67 | \( 1 + (-0.741 - 0.670i)T \) |
| 71 | \( 1 + (0.991 - 0.133i)T \) |
| 73 | \( 1 + (0.975 + 0.220i)T \) |
| 79 | \( 1 + (-0.420 + 0.907i)T \) |
| 83 | \( 1 + (0.0111 - 0.999i)T \) |
| 89 | \( 1 + (0.996 + 0.0890i)T \) |
| 97 | \( 1 + (-0.999 - 0.0222i)T \) |
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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
−25.69339582843534074859067694212, −24.79909782368649665128628348875, −24.27293090712024958831435549875, −23.09071325815107556855894695927, −21.63979193858272709242942143875, −20.77157327607197564064309014055, −19.99705945875859406007418484502, −19.04377050475772549435536541932, −18.434527925809321095912055620652, −17.828023311412776796009095336733, −16.11866183617921184540924046863, −15.53930653370866584587154899856, −14.756639599724994054879306331745, −13.77120171268798498987033350722, −12.24687303525039014549525695350, −11.360174383400629917280707810006, −10.443281804332413155110587603255, −9.257097112013790939629688545786, −8.18640366995657428478913480068, −7.87688368187651228124069547212, −6.75197175211177617755131190501, −5.30983560160321787676696587842, −3.309699142000808578725226808273, −2.76688057033483490066235670441, −1.22220036403211420587085935774,
1.23211277367397713747402402729, 2.33722842191079350599338471879, 3.87372129869427215128077899068, 4.727292333838649542969604324821, 6.85190222460632624449725304633, 7.6938465410885418942330972652, 8.39074476329100893538319928598, 9.24470855437894221617482535450, 10.28987814542317114922115598656, 11.286222718999741681882719779158, 12.34517091660779870357280202202, 13.47806398262308107410855577428, 14.7963164541052388567761046720, 15.4530900057298032708700097023, 16.401306957611901633636692499296, 17.314568340324577102370662793442, 18.37925534931608252415161685752, 19.434290842470254658045654960462, 19.93709512802389351747880371166, 20.87039515396532420130265583349, 21.2309455806776831570374289200, 23.12693516563420130135476280812, 24.27005795818642104541458593663, 24.54387869855113477957956156917, 25.94401619726084048802798766508