L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.939 − 0.342i)11-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + 17-s + (−0.5 + 0.866i)19-s + (0.173 + 0.984i)20-s + (0.766 + 0.642i)22-s + (0.766 + 0.642i)23-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.939 − 0.342i)11-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + 17-s + (−0.5 + 0.866i)19-s + (0.173 + 0.984i)20-s + (0.766 + 0.642i)22-s + (0.766 + 0.642i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8675434525 - 0.2028104009i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8675434525 - 0.2028104009i\) |
\(L(1)\) |
\(\approx\) |
\(0.7315484566 - 0.08198731491i\) |
\(L(1)\) |
\(\approx\) |
\(0.7315484566 - 0.08198731491i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 5 | \( 1 + (0.766 + 0.642i)T \) |
| 7 | \( 1 + (-0.939 - 0.342i)T \) |
| 11 | \( 1 + (-0.939 - 0.342i)T \) |
| 13 | \( 1 + (0.173 - 0.984i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.939 - 0.342i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.766 - 0.642i)T \) |
| 47 | \( 1 + (0.766 - 0.642i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.939 + 0.342i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (0.766 - 0.642i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 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
−22.06266802458014571464601966382, −21.13148664387172956149732110299, −20.6427300717271728835959272882, −19.62356697459887008569533692060, −18.85939346250970700446415764121, −18.270742931615842799174487535480, −17.33138545425852199681824245280, −16.52084832072088736728327725214, −16.15572500305507155433503331959, −15.13493220053351159889459235061, −14.27626582500249958652363075792, −13.10262766461184268306082422256, −12.586970089784598254667575144736, −11.38297760696064915010172805993, −10.44359608408376042684884074529, −9.557664739150084561965352815819, −9.18754727062488750651903639096, −8.236188379891783443797810604980, −7.17353538432938376495752597252, −6.32024898731673086507705974692, −5.58349128742684025757445868143, −4.61449466669156460031587423358, −2.88682654310237257972072659650, −2.11993745738328482556722684144, −0.88186999985175946826140427283,
0.7321786754074759796574231409, 2.09103119192730559633166492615, 3.04893317820333388404840270192, 3.61215546009945444963735976022, 5.6003831469648849027101812140, 6.124553266051859227656850509724, 7.325448648079944153743912883328, 7.83558341215842327451932619814, 9.086514554929881837005361250853, 9.8577862854023823075950700553, 10.4984661073114479586854661867, 10.98650031213678185868980585899, 12.39232212334508304911082999584, 12.99724350611176514988114608433, 13.82190979926564630781155523161, 15.0784985264283552475141842577, 15.75338093386969605905841838391, 16.80994362441732497258504398820, 17.2075804440721457804153885753, 18.38966319814660473273004777439, 18.696870465660502334225544124887, 19.49281371563627282053152180905, 20.57639023825584847215511455503, 21.05096019749993890985202495322, 21.93644855432106249051074030159