L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.642 − 0.766i)3-s + (0.173 + 0.984i)4-s − i·6-s + (−0.342 + 0.939i)7-s + (−0.5 + 0.866i)8-s + (−0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (0.642 − 0.766i)12-s + (0.173 + 0.984i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + (−0.766 + 0.642i)18-s + (0.642 + 0.766i)19-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.642 − 0.766i)3-s + (0.173 + 0.984i)4-s − i·6-s + (−0.342 + 0.939i)7-s + (−0.5 + 0.866i)8-s + (−0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (0.642 − 0.766i)12-s + (0.173 + 0.984i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + (−0.766 + 0.642i)18-s + (0.642 + 0.766i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0103 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0103 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8968177332 + 0.9061735284i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8968177332 + 0.9061735284i\) |
\(L(1)\) |
\(\approx\) |
\(1.099624429 + 0.5112220127i\) |
\(L(1)\) |
\(\approx\) |
\(1.099624429 + 0.5112220127i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.642 - 0.766i)T \) |
| 7 | \( 1 + (-0.342 + 0.939i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.173 + 0.984i)T \) |
| 17 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 + (0.642 + 0.766i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.866 + 0.5i)T \) |
| 31 | \( 1 + iT \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.342 - 0.939i)T \) |
| 59 | \( 1 + (-0.342 - 0.939i)T \) |
| 61 | \( 1 + (-0.984 + 0.173i)T \) |
| 67 | \( 1 + (0.342 - 0.939i)T \) |
| 71 | \( 1 + (0.766 - 0.642i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.342 - 0.939i)T \) |
| 83 | \( 1 + (0.984 + 0.173i)T \) |
| 89 | \( 1 + (0.342 + 0.939i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−27.34631401756622203636743962015, −26.177219832835390158836422716182, −24.89244370899192077233707936136, −23.66420284863541440875832999919, −22.80655310981307053340082962598, −22.44475491039242922937837200588, −21.28574494066075428505200478171, −20.20977630370112985555939551889, −19.90526234724114834384395133513, −18.15428196191684762334384928364, −17.27967143455232158334891686308, −15.95261739115853721063864188743, −15.28262951339039654288342626757, −14.06897268476230534144642903281, −13.084274659994519923057302160999, −11.91975039836300068282006318728, −11.11552519083440678389390059403, −10.02250102681113400085813769165, −9.492662806407260524134567855382, −7.27052999867788355400977673478, −6.12608362768173226485913473397, −4.930594381427675366563515059587, −4.08158382231758161585610912717, −2.95524873132855922139063865006, −0.8998265175637181612480867120,
1.93560552613477480697978842388, 3.43797658143237943996845583008, 4.949089860249297396000289009986, 6.149225305877684737744892950473, 6.51429514410763977746586250302, 8.01896863360708121073134450590, 8.91733170965552771315870191075, 10.88845448378333670225095229129, 12.03904907422330312767745584069, 12.4654007964860193067370983966, 13.74815453455075846391308196396, 14.47644642191362978283564295816, 15.99546739070922681764694583394, 16.492873159569965512100149375732, 17.64115144246627134490345996653, 18.63412733781783412232071329914, 19.54783354400820665413066222728, 21.225365744460881810928492388144, 21.99904092688421259276886824276, 22.68987097841737738344884114761, 23.83577717334997196545088166123, 24.40818016864218535104000820933, 25.222510289490799015870545529971, 26.19105890857961918751411812695, 27.42606143650736492273077092094