L(s) = 1 | + (0.642 − 0.766i)2-s + (0.642 + 0.766i)3-s + (−0.173 − 0.984i)4-s + 6-s + (−0.342 + 0.939i)7-s + (−0.866 − 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.642 − 0.766i)12-s + (−0.984 + 0.173i)13-s + (0.5 + 0.866i)14-s + (−0.939 + 0.342i)16-s + (−0.984 − 0.173i)17-s + (0.642 + 0.766i)18-s + (−0.766 + 0.642i)19-s + ⋯ |
L(s) = 1 | + (0.642 − 0.766i)2-s + (0.642 + 0.766i)3-s + (−0.173 − 0.984i)4-s + 6-s + (−0.342 + 0.939i)7-s + (−0.866 − 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.642 − 0.766i)12-s + (−0.984 + 0.173i)13-s + (0.5 + 0.866i)14-s + (−0.939 + 0.342i)16-s + (−0.984 − 0.173i)17-s + (0.642 + 0.766i)18-s + (−0.766 + 0.642i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9797995095 + 1.201163531i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9797995095 + 1.201163531i\) |
\(L(1)\) |
\(\approx\) |
\(1.308961728 + 0.1048610502i\) |
\(L(1)\) |
\(\approx\) |
\(1.308961728 + 0.1048610502i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (0.642 - 0.766i)T \) |
| 3 | \( 1 + (0.642 + 0.766i)T \) |
| 7 | \( 1 + (-0.342 + 0.939i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.984 + 0.173i)T \) |
| 17 | \( 1 + (-0.984 - 0.173i)T \) |
| 19 | \( 1 + (-0.766 + 0.642i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 + (0.173 + 0.984i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.342 - 0.939i)T \) |
| 59 | \( 1 + (0.939 - 0.342i)T \) |
| 61 | \( 1 + (0.173 + 0.984i)T \) |
| 67 | \( 1 + (-0.342 + 0.939i)T \) |
| 71 | \( 1 + (0.766 - 0.642i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.984 + 0.173i)T \) |
| 89 | \( 1 + (0.939 - 0.342i)T \) |
| 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
−26.4660553091651249429622378889, −25.67157730916098639640730054135, −24.63722243360737282550394802306, −23.928144228681718457903094150090, −23.25567193973119543721074853234, −22.06041007766337764661537260247, −21.03294384414340419603372341970, −19.92964776928835917795682799179, −19.08537575477435365388323696339, −17.68412991941189674971448325311, −17.04200736120043100410348189973, −15.74721904143321098795182401096, −14.81568974469110076690380549978, −13.71280831940048807056880863259, −13.25141517210584701041307762916, −12.260959114871516673006679189383, −10.829137452598917681627410093406, −9.14812566263945200904793816530, −8.16088236865944724843589311605, −7.13930795865916150619748806958, −6.45812086588898602316542860768, −4.9250892345172724686786149946, −3.58002023097552959559961215301, −2.5324980706493468567520918680, −0.36855191191227429901588307713,
2.20286208327197564877207504257, 2.80232936601443759980097280616, 4.363340953928829503993430850291, 5.04837230702422207001762436164, 6.51834641073883496746280393858, 8.31420400265645015463027209830, 9.49435879551895382091838236845, 10.09073257664500922404431912320, 11.3290740543603655701438124859, 12.48626713482707159826531787844, 13.30907403323040671519466765621, 14.70766594556399312575807134088, 15.097548082401443913861473046401, 16.12385046663489087760622473894, 17.730508145854340941217032219509, 19.057438786476950185959952143159, 19.61347755382162266962744345098, 20.774691949867254025163933299319, 21.35778463656999212857397912599, 22.33744430878165085946173295450, 22.97975711635782088734151058875, 24.52894868043645744965360801490, 25.18313110568513106181301908654, 26.4569235506809528459007506338, 27.35991143231285835584255913397