L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.766 − 0.642i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.766 + 0.642i)10-s − 11-s + (−0.766 + 0.642i)13-s + (0.766 + 0.642i)14-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + (0.5 + 0.866i)20-s + (−0.173 + 0.984i)22-s + (0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s + (0.5 + 0.866i)26-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.766 − 0.642i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.766 + 0.642i)10-s − 11-s + (−0.766 + 0.642i)13-s + (0.766 + 0.642i)14-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + (0.5 + 0.866i)20-s + (−0.173 + 0.984i)22-s + (0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s + (0.5 + 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01455150485 + 0.01832694323i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01455150485 + 0.01832694323i\) |
\(L(1)\) |
\(\approx\) |
\(0.5110222843 - 0.2656830117i\) |
\(L(1)\) |
\(\approx\) |
\(0.5110222843 - 0.2656830117i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.766 + 0.642i)T \) |
| 17 | \( 1 + (-0.766 - 0.642i)T \) |
| 23 | \( 1 + (0.939 + 0.342i)T \) |
| 29 | \( 1 + (-0.939 - 0.342i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.939 + 0.342i)T \) |
| 47 | \( 1 + (0.939 + 0.342i)T \) |
| 53 | \( 1 + (0.173 + 0.984i)T \) |
| 59 | \( 1 + (-0.939 + 0.342i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (-0.173 - 0.984i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + (-0.939 + 0.342i)T \) |
| 79 | \( 1 + (-0.766 - 0.642i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.939 - 0.342i)T \) |
| 97 | \( 1 + (-0.173 + 0.984i)T \) |
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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
−26.74878672653763613841863293229, −26.48039460707851021125847291057, −25.45559093668771287180648141108, −24.17460041298007554124454931516, −23.470697282550491725334243727177, −22.685260816839054674451337845903, −21.896113542944813699775813191689, −20.36458753425107111930173893674, −19.29495330140177333629890459796, −18.33198243027128368976614583304, −17.27147027191491839917399821335, −16.29496793345671805512629021129, −15.32508689103626847276544413913, −14.64967293524355607273705741442, −13.32170984674612507967025198400, −12.620385346852594375404906755939, −10.930039825067131697696837803882, −10.00859092483705752492459599613, −8.48740987329539334033155844061, −7.42136929746255472847332929186, −6.83785615973596732647962800084, −5.34865453579545785341735487973, −4.1003131735355063957578918889, −3.02301842551015574964946845090, −0.01714938713934720387468219815,
2.078510168165918908656994920799, 3.26239840953806577422403015056, 4.64336724758746083563458852264, 5.49108280160500008965229533665, 7.38319461634831683148877664026, 8.80955304032580355455157711522, 9.4181799005848297581331159468, 10.87475020179308325381429396789, 11.8557068493515204239088568206, 12.63207888010854471069133601326, 13.47834079960953107961301854862, 14.97851251004152654002668759216, 15.82440573443935772337368297338, 17.10410659768018314827588344278, 18.48123528451503606061038748623, 19.10362277851848633862774025559, 20.069260484024503600835642333398, 20.95689639562489173560992869036, 21.91773332440046401021563368997, 22.84600566755666685664525467008, 23.81264418828852934856524860186, 24.68235691207657816500254357136, 26.18509511533228785346471555080, 27.09202123732363173534321426982, 28.02493185052929374759599068068