L(s) = 1 | + (0.483 + 0.875i)2-s + (−0.532 + 0.846i)4-s + (0.935 + 0.353i)5-s + (−0.380 − 0.924i)7-s + (−0.998 − 0.0570i)8-s + (0.142 + 0.989i)10-s + (−0.640 − 0.768i)13-s + (0.625 − 0.780i)14-s + (−0.432 − 0.901i)16-s + (0.993 − 0.113i)17-s + (0.870 − 0.491i)19-s + (−0.797 + 0.603i)20-s + (−0.235 − 0.971i)23-s + (0.749 + 0.662i)25-s + (0.362 − 0.931i)26-s + ⋯ |
L(s) = 1 | + (0.483 + 0.875i)2-s + (−0.532 + 0.846i)4-s + (0.935 + 0.353i)5-s + (−0.380 − 0.924i)7-s + (−0.998 − 0.0570i)8-s + (0.142 + 0.989i)10-s + (−0.640 − 0.768i)13-s + (0.625 − 0.780i)14-s + (−0.432 − 0.901i)16-s + (0.993 − 0.113i)17-s + (0.870 − 0.491i)19-s + (−0.797 + 0.603i)20-s + (−0.235 − 0.971i)23-s + (0.749 + 0.662i)25-s + (0.362 − 0.931i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.884945558 + 0.3166896050i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.884945558 + 0.3166896050i\) |
\(L(1)\) |
\(\approx\) |
\(1.317366266 + 0.4376584125i\) |
\(L(1)\) |
\(\approx\) |
\(1.317366266 + 0.4376584125i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.483 + 0.875i)T \) |
| 5 | \( 1 + (0.935 + 0.353i)T \) |
| 7 | \( 1 + (-0.380 - 0.924i)T \) |
| 13 | \( 1 + (-0.640 - 0.768i)T \) |
| 17 | \( 1 + (0.993 - 0.113i)T \) |
| 19 | \( 1 + (0.870 - 0.491i)T \) |
| 23 | \( 1 + (-0.235 - 0.971i)T \) |
| 29 | \( 1 + (-0.948 - 0.318i)T \) |
| 31 | \( 1 + (-0.830 - 0.556i)T \) |
| 37 | \( 1 + (0.696 + 0.717i)T \) |
| 41 | \( 1 + (0.820 - 0.572i)T \) |
| 43 | \( 1 + (-0.0475 - 0.998i)T \) |
| 47 | \( 1 + (0.290 - 0.956i)T \) |
| 53 | \( 1 + (0.564 + 0.825i)T \) |
| 59 | \( 1 + (0.905 - 0.424i)T \) |
| 61 | \( 1 + (0.999 - 0.0190i)T \) |
| 67 | \( 1 + (-0.327 + 0.945i)T \) |
| 71 | \( 1 + (0.736 + 0.676i)T \) |
| 73 | \( 1 + (-0.941 + 0.336i)T \) |
| 79 | \( 1 + (0.710 + 0.703i)T \) |
| 83 | \( 1 + (-0.851 - 0.524i)T \) |
| 89 | \( 1 + (-0.415 - 0.909i)T \) |
| 97 | \( 1 + (-0.935 + 0.353i)T \) |
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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
−21.32637718622977496797439456819, −20.927144485198607838852153946540, −19.83252980104858559275158084917, −19.235265459579576895248340186418, −18.31295415894032181963796156845, −17.8783598773180066620111648571, −16.6629341300885125989339698523, −15.99589393650529777584859042809, −14.64105481661417855771976630947, −14.411743630615310630675599285741, −13.33966620792046369745634265654, −12.677325162134430638326957208967, −12.031694269347111166747612150929, −11.260568432305100134754894426314, −10.09985284238851915312281004969, −9.43542826876372877337192555590, −9.12552527234349055375001038726, −7.740369221597713002620139510775, −6.406068148485015240785974472141, −5.55411922598370513196120220259, −5.1882952334835199227199081661, −3.89696307916516939333211804986, −2.93522155827004727042033402848, −2.0577415153659730096853779003, −1.268402317527560735387318921120,
0.71792907866883491090157376811, 2.43777037626499236536933735914, 3.29305343568110446315451920041, 4.23278783786026393033938013931, 5.40807407901112265348433916581, 5.820661901299258775129089497979, 7.08008999761424199193756993303, 7.31624168572123118223211664958, 8.459099542825917664635300612229, 9.6438661878743686988048138732, 10.02159349897310782017049137701, 11.1532406896311354463531661901, 12.34643287812197475893019955232, 13.08093872229261242330471492620, 13.70315625929473613116644083448, 14.4377992615985127975126535420, 15.02960751215354702311530387551, 16.13181742095267554811160032591, 16.84578542218391940367411638665, 17.32930243737946047864709824200, 18.19256778292649449444548180476, 18.88364794505861151559922816635, 20.28289447517120840258479544657, 20.66415149802123638578582983677, 21.83160982678045318765671611083