L(s) = 1 | + (0.342 + 0.939i)3-s + (−0.984 − 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.642 − 0.766i)13-s + (−0.642 − 0.766i)17-s + (−0.939 + 0.342i)19-s + (−0.173 − 0.984i)21-s + (0.866 + 0.5i)23-s + (−0.866 − 0.5i)27-s + (0.5 + 0.866i)29-s + 31-s + (0.642 − 0.766i)33-s + (0.939 + 0.342i)39-s + (0.766 + 0.642i)41-s + ⋯ |
L(s) = 1 | + (0.342 + 0.939i)3-s + (−0.984 − 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.642 − 0.766i)13-s + (−0.642 − 0.766i)17-s + (−0.939 + 0.342i)19-s + (−0.173 − 0.984i)21-s + (0.866 + 0.5i)23-s + (−0.866 − 0.5i)27-s + (0.5 + 0.866i)29-s + 31-s + (0.642 − 0.766i)33-s + (0.939 + 0.342i)39-s + (0.766 + 0.642i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.076083187 + 0.7132193756i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.076083187 + 0.7132193756i\) |
\(L(1)\) |
\(\approx\) |
\(0.9682492995 + 0.2787968902i\) |
\(L(1)\) |
\(\approx\) |
\(0.9682492995 + 0.2787968902i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (0.342 + 0.939i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.642 - 0.766i)T \) |
| 17 | \( 1 + (-0.642 - 0.766i)T \) |
| 19 | \( 1 + (-0.939 + 0.342i)T \) |
| 23 | \( 1 + (0.866 + 0.5i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.984 + 0.173i)T \) |
| 59 | \( 1 + (0.173 + 0.984i)T \) |
| 61 | \( 1 + (0.766 + 0.642i)T \) |
| 67 | \( 1 + (0.984 + 0.173i)T \) |
| 71 | \( 1 + (0.939 - 0.342i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (-0.173 + 0.984i)T \) |
| 83 | \( 1 + (-0.642 - 0.766i)T \) |
| 89 | \( 1 + (0.173 + 0.984i)T \) |
| 97 | \( 1 + (0.866 + 0.5i)T \) |
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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
−20.434674576100937480813941243038, −19.58319136254483680671764572199, −18.98129183777092603912947677844, −18.566302242968906117664421790444, −17.37416876553360107567280294134, −17.13404000016700749593446349203, −15.74979523705047913777281467488, −15.36850223144347758251951288882, −14.345881408712257012381025885695, −13.535900466703238262897431458252, −12.83931302869353385820856871387, −12.470247142873842883923729758837, −11.44434308230917544353762923773, −10.55111807996500791825092729921, −9.59098460085983401526268351576, −8.77496877105062098667542903189, −8.19323439532842175326938657298, −6.964173354098037693431378994252, −6.63742015395049171693723667013, −5.836385281378811856898036217937, −4.53140874528476146745381827925, −3.644128890073470843984602740891, −2.497170096940057256696570974, −2.021792744537163481400760908848, −0.606004767064991741779778439548,
0.85641619965824932750672895763, 2.66446217745441923818062701858, 3.07595333020247708310103892069, 3.97806953290695996417569950980, 4.91088945346105086297298044182, 5.81797100613046702721869421883, 6.548332683665676672778529903031, 7.75322970252957543126405152560, 8.57208662193996696865158208095, 9.20456711326249634833287771943, 10.06763452889828883825685439871, 10.76838175253419549352106693643, 11.29538805501724331126914225752, 12.62035725505126119396262646308, 13.32390381451327808127776072184, 13.913502860182783466666715569621, 14.90749026353232315306820799540, 15.74783071307923173717259650114, 16.04779656646765707322061283851, 16.84349752045931855084562600532, 17.71927941328432835342991267804, 18.735855328408228962511887492733, 19.42721668406760279283003255567, 20.055007127218295844138917425042, 20.91565617263993399414800266448