L(s) = 1 | − i·2-s + (−0.893 + 0.448i)3-s − 4-s + (−0.116 − 0.993i)5-s + (0.448 + 0.893i)6-s + (0.597 + 0.802i)7-s + i·8-s + (0.597 − 0.802i)9-s + (−0.993 + 0.116i)10-s + (0.993 + 0.116i)11-s + (0.893 − 0.448i)12-s + (−0.116 − 0.993i)13-s + (0.802 − 0.597i)14-s + (0.549 + 0.835i)15-s + 16-s + (0.866 − 0.5i)17-s + ⋯ |
L(s) = 1 | − i·2-s + (−0.893 + 0.448i)3-s − 4-s + (−0.116 − 0.993i)5-s + (0.448 + 0.893i)6-s + (0.597 + 0.802i)7-s + i·8-s + (0.597 − 0.802i)9-s + (−0.993 + 0.116i)10-s + (0.993 + 0.116i)11-s + (0.893 − 0.448i)12-s + (−0.116 − 0.993i)13-s + (0.802 − 0.597i)14-s + (0.549 + 0.835i)15-s + 16-s + (0.866 − 0.5i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01406189475 + 0.01586218538i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01406189475 + 0.01586218538i\) |
\(L(1)\) |
\(\approx\) |
\(0.6598346476 - 0.3586095432i\) |
\(L(1)\) |
\(\approx\) |
\(0.6598346476 - 0.3586095432i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.893 + 0.448i)T \) |
| 5 | \( 1 + (-0.116 - 0.993i)T \) |
| 7 | \( 1 + (0.597 + 0.802i)T \) |
| 11 | \( 1 + (0.993 + 0.116i)T \) |
| 13 | \( 1 + (-0.116 - 0.993i)T \) |
| 17 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.984 + 0.173i)T \) |
| 23 | \( 1 + (0.642 + 0.766i)T \) |
| 29 | \( 1 + (0.802 - 0.597i)T \) |
| 31 | \( 1 + (-0.230 - 0.973i)T \) |
| 41 | \( 1 + (-0.766 - 0.642i)T \) |
| 43 | \( 1 + (0.984 + 0.173i)T \) |
| 47 | \( 1 + (0.893 + 0.448i)T \) |
| 53 | \( 1 + (-0.0581 + 0.998i)T \) |
| 59 | \( 1 + (-0.549 - 0.835i)T \) |
| 61 | \( 1 + (-0.957 + 0.286i)T \) |
| 67 | \( 1 + (-0.893 - 0.448i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.597 + 0.802i)T \) |
| 79 | \( 1 + (-0.549 - 0.835i)T \) |
| 83 | \( 1 + (-0.686 + 0.727i)T \) |
| 89 | \( 1 + (-0.918 - 0.396i)T \) |
| 97 | \( 1 + (-0.230 + 0.973i)T \) |
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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
−18.60663936520087880181102965699, −17.95576501432703683315643631794, −17.31282263080207319446166700326, −16.76509125613602516502432454397, −16.43071818348367878826998767530, −15.33819287836704193712310507596, −14.60648894465864979429696207597, −14.17476058664431002168648692622, −13.64107627614311023677704329654, −12.603083849237647019332873013, −12.023857881181272866646639668984, −11.13924155179333319187902675365, −10.55812147754155127605683871313, −9.97210250392686551359789665886, −8.84156920929053947386043366755, −8.165546419863273900670078865317, −7.190092720727614781108323883454, −6.92084361459038327117522026362, −6.37696841637609763632673233075, −5.61079964610217563185824878945, −4.52653516994632762200013640426, −4.23533648499906411163215358205, −3.199882101421999218275684536301, −1.75942803862094021558921647638, −1.06830182924264063458539310656,
0.004560694247810970017119876531, 0.940586631175613342830226819627, 1.42399222777040579699831564027, 2.51711724297641617873327927121, 3.55883073128706234359694396302, 4.32056120915443639825536919904, 4.859568344230850792872936075912, 5.61399955115890557149611756102, 6.0135778997977493868034713922, 7.50610716952958010314650401968, 8.25076948972710622089561767604, 9.12809653266104122993103435903, 9.42588428464001013331386758954, 10.30219285994849697823630374983, 11.04727602443048109361638257026, 11.71142736621868765634903741344, 12.25698578302001159104510454932, 12.54866022384923846108451924952, 13.462413280097936161874517623612, 14.39248390038766816707168311026, 15.17140827538871481370042807438, 15.68798241619143963780634838541, 16.80726693492645899869491547083, 17.27314104681980512069173169526, 17.58282534070875075284283402456