L(s) = 1 | + (0.156 + 0.987i)2-s + (0.987 + 0.156i)3-s + (−0.951 + 0.309i)4-s + i·6-s + (−0.522 − 0.852i)7-s + (−0.453 − 0.891i)8-s + (0.951 + 0.309i)9-s + (0.852 − 0.522i)11-s + (−0.987 + 0.156i)12-s + (−0.972 − 0.233i)13-s + (0.760 − 0.649i)14-s + (0.809 − 0.587i)16-s + (0.522 − 0.852i)17-s + (−0.156 + 0.987i)18-s + (−0.649 + 0.760i)19-s + ⋯ |
L(s) = 1 | + (0.156 + 0.987i)2-s + (0.987 + 0.156i)3-s + (−0.951 + 0.309i)4-s + i·6-s + (−0.522 − 0.852i)7-s + (−0.453 − 0.891i)8-s + (0.951 + 0.309i)9-s + (0.852 − 0.522i)11-s + (−0.987 + 0.156i)12-s + (−0.972 − 0.233i)13-s + (0.760 − 0.649i)14-s + (0.809 − 0.587i)16-s + (0.522 − 0.852i)17-s + (−0.156 + 0.987i)18-s + (−0.649 + 0.760i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.283107274 - 0.03868754873i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.283107274 - 0.03868754873i\) |
\(L(1)\) |
\(\approx\) |
\(1.332441703 + 0.4282980509i\) |
\(L(1)\) |
\(\approx\) |
\(1.332441703 + 0.4282980509i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (0.156 + 0.987i)T \) |
| 3 | \( 1 + (0.987 + 0.156i)T \) |
| 7 | \( 1 + (-0.522 - 0.852i)T \) |
| 11 | \( 1 + (0.852 - 0.522i)T \) |
| 13 | \( 1 + (-0.972 - 0.233i)T \) |
| 17 | \( 1 + (0.522 - 0.852i)T \) |
| 19 | \( 1 + (-0.649 + 0.760i)T \) |
| 23 | \( 1 + (0.649 - 0.760i)T \) |
| 29 | \( 1 + (0.156 + 0.987i)T \) |
| 31 | \( 1 + (0.852 + 0.522i)T \) |
| 37 | \( 1 + (0.0784 - 0.996i)T \) |
| 41 | \( 1 + (0.707 - 0.707i)T \) |
| 43 | \( 1 + (0.852 + 0.522i)T \) |
| 47 | \( 1 + (-0.987 + 0.156i)T \) |
| 53 | \( 1 + (0.156 + 0.987i)T \) |
| 59 | \( 1 + (-0.891 - 0.453i)T \) |
| 61 | \( 1 + (0.891 - 0.453i)T \) |
| 67 | \( 1 + (-0.987 - 0.156i)T \) |
| 71 | \( 1 + (0.233 - 0.972i)T \) |
| 73 | \( 1 + (-0.972 + 0.233i)T \) |
| 79 | \( 1 + (0.707 - 0.707i)T \) |
| 83 | \( 1 + (-0.809 - 0.587i)T \) |
| 89 | \( 1 + (0.233 - 0.972i)T \) |
| 97 | \( 1 + (-0.809 - 0.587i)T \) |
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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
−17.67802336356940486793830625037, −17.4281098577061497142796177398, −16.44320965526878930070390110219, −15.16184720878061316206351757786, −15.10853440073143254994687151702, −14.48617906455061224936991696006, −13.55630078716189296170290601439, −13.11653242201675090707343647335, −12.39543984529184393290996014704, −11.98635563344625585473883621089, −11.26389434434096788223030068473, −10.1608372416881078391361258897, −9.64896968275396767536479172310, −9.3097092365973486149030732738, −8.5214285298277756938239465537, −7.948520788792349922963677438241, −6.91301628026097116249412001025, −6.25979272468362291859311025806, −5.294055306995096597333781325702, −4.3970800908691300189433895360, −3.95008085805057980237873659585, −2.954971674657317438752987025807, −2.54913119653088604697776049012, −1.8276152102039073749983806328, −1.026523598430288848573802945896,
0.52175997817626880899380911207, 1.455919788751763742494147545111, 2.88238779286647495580221275562, 3.222039771388302191930910962979, 4.20143313593007788935263345270, 4.53456233854826140570360842346, 5.51950278113311689270243579908, 6.46326790000692386639221817208, 7.01021664827515474407774436061, 7.59201542900345924636039744736, 8.21315265454712392456041544846, 9.0861602949394236391035694859, 9.40990661250285305297661074580, 10.21643386127182452596353957149, 10.80003479730731997284929891641, 12.23405036121711288451347343915, 12.573537619076162342973583301967, 13.362366690617933838165679383270, 14.07830648029504569137167632832, 14.44201016211494346491471456617, 14.82420395793434845771970166100, 15.92665728848916178612699517461, 16.242286855415330983059249821795, 16.87976227912012205126549599048, 17.47911727099058421111234166078