L(s) = 1 | + (−0.222 − 0.974i)2-s + (0.955 − 0.294i)3-s + (−0.900 + 0.433i)4-s + (0.365 − 0.930i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (0.623 + 0.781i)8-s + (0.826 − 0.563i)9-s + (−0.988 − 0.149i)10-s + (−0.900 − 0.433i)11-s + (−0.733 + 0.680i)12-s + (−0.988 + 0.149i)13-s + (0.955 + 0.294i)14-s + (0.0747 − 0.997i)15-s + (0.623 − 0.781i)16-s + (0.365 + 0.930i)17-s + ⋯ |
L(s) = 1 | + (−0.222 − 0.974i)2-s + (0.955 − 0.294i)3-s + (−0.900 + 0.433i)4-s + (0.365 − 0.930i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (0.623 + 0.781i)8-s + (0.826 − 0.563i)9-s + (−0.988 − 0.149i)10-s + (−0.900 − 0.433i)11-s + (−0.733 + 0.680i)12-s + (−0.988 + 0.149i)13-s + (0.955 + 0.294i)14-s + (0.0747 − 0.997i)15-s + (0.623 − 0.781i)16-s + (0.365 + 0.930i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0689 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0689 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6494733595 - 0.6061340416i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6494733595 - 0.6061340416i\) |
\(L(1)\) |
\(\approx\) |
\(0.8840804779 - 0.5570339266i\) |
\(L(1)\) |
\(\approx\) |
\(0.8840804779 - 0.5570339266i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (-0.222 - 0.974i)T \) |
| 3 | \( 1 + (0.955 - 0.294i)T \) |
| 5 | \( 1 + (0.365 - 0.930i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.900 - 0.433i)T \) |
| 13 | \( 1 + (-0.988 + 0.149i)T \) |
| 17 | \( 1 + (0.365 + 0.930i)T \) |
| 19 | \( 1 + (0.826 + 0.563i)T \) |
| 23 | \( 1 + (0.0747 + 0.997i)T \) |
| 29 | \( 1 + (0.955 + 0.294i)T \) |
| 31 | \( 1 + (-0.733 + 0.680i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.222 - 0.974i)T \) |
| 47 | \( 1 + (-0.900 + 0.433i)T \) |
| 53 | \( 1 + (-0.988 - 0.149i)T \) |
| 59 | \( 1 + (0.623 - 0.781i)T \) |
| 61 | \( 1 + (-0.733 - 0.680i)T \) |
| 67 | \( 1 + (0.826 + 0.563i)T \) |
| 71 | \( 1 + (0.0747 - 0.997i)T \) |
| 73 | \( 1 + (-0.988 + 0.149i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.955 - 0.294i)T \) |
| 89 | \( 1 + (0.955 - 0.294i)T \) |
| 97 | \( 1 + (-0.900 - 0.433i)T \) |
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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
−34.743177269842302173390371484376, −33.60749162493852179142749689309, −32.78123595018292607149697730052, −31.669960729134348572113021300050, −30.5337826387271693403400679518, −29.08159497846115804865423718746, −27.16382049165488375404437231824, −26.43444922253288544490396421459, −25.73054907281858974199303257266, −24.55993959409408064107712036163, −23.05184130776454648038264636272, −22.0201033628661463292317480403, −20.31562046889861429122162685200, −19.02274130929544205458238444316, −17.91874482058518616747235330299, −16.372180496884025786348018572372, −15.17140128180055920415883080945, −14.154395654900171166207821086032, −13.22718222550756434011132963520, −10.281008694734744769443212737, −9.63940247179466660017246297637, −7.73518149505279361901558572891, −6.917270166114516528455614197966, −4.80975307769848090140862527223, −2.93728946461447657819054500970,
1.86260340148840904965211612844, 3.29713646278602387968652393280, 5.27880869697051129252511302849, 7.94832510952212288593065451181, 9.04407987196600115838848901355, 10.00855768219029390414381628208, 12.23551693559762602526863768606, 12.90853069925690055644963294595, 14.18268969754224497370229540739, 15.98840629311402676778799692859, 17.69713717739764711820591696888, 18.96576016876800751162594305616, 19.81431887888280111434649104850, 21.069918258973434878781324514726, 21.782143920515970964666377610880, 23.76076005685048328672399913772, 25.06635367429451204434867378846, 26.13352204946004327089705011453, 27.41872210335725215569764291478, 28.81062281480994805516433735998, 29.425542948752854178789956300070, 31.07008018141388643286914471576, 31.72250825152723560211843516363, 32.566591500663159248576126307793, 34.83082789509458443117185264804