L(s) = 1 | + (0.0334 + 0.999i)2-s + (0.420 − 0.907i)3-s + (−0.997 + 0.0667i)4-s + (0.645 − 0.763i)5-s + (0.920 + 0.390i)6-s + (−0.166 − 0.986i)7-s + (−0.100 − 0.994i)8-s + (−0.645 − 0.763i)9-s + (0.784 + 0.619i)10-s + (−0.997 − 0.0667i)11-s + (−0.359 + 0.933i)12-s + (0.920 + 0.390i)13-s + (0.979 − 0.199i)14-s + (−0.420 − 0.907i)15-s + (0.991 − 0.133i)16-s + (0.979 + 0.199i)17-s + ⋯ |
L(s) = 1 | + (0.0334 + 0.999i)2-s + (0.420 − 0.907i)3-s + (−0.997 + 0.0667i)4-s + (0.645 − 0.763i)5-s + (0.920 + 0.390i)6-s + (−0.166 − 0.986i)7-s + (−0.100 − 0.994i)8-s + (−0.645 − 0.763i)9-s + (0.784 + 0.619i)10-s + (−0.997 − 0.0667i)11-s + (−0.359 + 0.933i)12-s + (0.920 + 0.390i)13-s + (0.979 − 0.199i)14-s + (−0.420 − 0.907i)15-s + (0.991 − 0.133i)16-s + (0.979 + 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6097429997 - 1.228105329i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6097429997 - 1.228105329i\) |
\(L(1)\) |
\(\approx\) |
\(1.040332379 - 0.2319563204i\) |
\(L(1)\) |
\(\approx\) |
\(1.040332379 - 0.2319563204i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 283 | \( 1 \) |
good | 2 | \( 1 + (0.0334 + 0.999i)T \) |
| 3 | \( 1 + (0.420 - 0.907i)T \) |
| 5 | \( 1 + (0.645 - 0.763i)T \) |
| 7 | \( 1 + (-0.166 - 0.986i)T \) |
| 11 | \( 1 + (-0.997 - 0.0667i)T \) |
| 13 | \( 1 + (0.920 + 0.390i)T \) |
| 17 | \( 1 + (0.979 + 0.199i)T \) |
| 19 | \( 1 + (-0.920 - 0.390i)T \) |
| 23 | \( 1 + (0.860 - 0.509i)T \) |
| 29 | \( 1 + (-0.741 - 0.670i)T \) |
| 31 | \( 1 + (-0.593 + 0.805i)T \) |
| 37 | \( 1 + (-0.480 + 0.876i)T \) |
| 41 | \( 1 + (0.100 - 0.994i)T \) |
| 43 | \( 1 + (0.538 + 0.842i)T \) |
| 47 | \( 1 + (-0.920 + 0.390i)T \) |
| 53 | \( 1 + (-0.991 - 0.133i)T \) |
| 59 | \( 1 + (-0.944 + 0.328i)T \) |
| 61 | \( 1 + (0.784 - 0.619i)T \) |
| 67 | \( 1 + (-0.359 - 0.933i)T \) |
| 71 | \( 1 + (-0.997 - 0.0667i)T \) |
| 73 | \( 1 + (0.593 + 0.805i)T \) |
| 79 | \( 1 + (0.538 - 0.842i)T \) |
| 83 | \( 1 + (0.964 - 0.264i)T \) |
| 89 | \( 1 + (-0.538 - 0.842i)T \) |
| 97 | \( 1 + (0.860 - 0.509i)T \) |
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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
−25.74996640106808679596412043959, −25.29743743057322267571994199103, −23.39925785934863535901853729138, −22.642714825542750828517226470856, −21.79357298314384959767232394566, −21.08804154991076406764217236103, −20.663393554990239379047286623749, −19.18218095141009032685843781897, −18.64036884832704609186365950380, −17.76153929624183450721859443199, −16.464194040697164696063529639094, −15.19608647609799850564630111284, −14.62225314577935086526598884633, −13.491741660979428018074896746098, −12.699239437561013116739493084084, −11.215147058514524437981256831536, −10.62994951776608285993256914861, −9.72589484773727255434204244030, −8.946525331199906370412171497708, −7.88488841828978351031603715160, −5.80610400053617907011759853802, −5.2233814619674328079174136924, −3.603541275987332428332604711019, −2.890993615134830737234008116367, −1.92292782596410760063713498888,
0.38636921516048829911181002429, 1.51219554178917384064892466709, 3.32387337448903001781790869015, 4.66989292446419641573924584258, 5.86014514664053237738077484936, 6.710036750830742043453378076737, 7.775520257421108672814396119590, 8.53195358891891606734892466661, 9.48632864237857297175532547754, 10.71910406509610169890419089858, 12.57493955766284386896502223507, 13.148812068536527328826772495878, 13.774787780803511107058031781130, 14.65806900128168376768310153481, 15.970636103537255495845225094207, 16.87241317957701533545778871709, 17.51401743524145977877384109312, 18.52685581713897282467865755733, 19.292012905485260399538965203696, 20.66774287064400271226700320111, 21.19134145984551625369747109726, 22.84415731147892390046780464399, 23.62275044734765977852605545325, 24.02207400839697959450720737508, 25.08769511056921758676348936915