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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.009190992 + 0.2327158741i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.009190992 + 0.2327158741i\) |
\(L(1)\) |
\(\approx\) |
\(0.9511875000 + 0.1379616514i\) |
\(L(1)\) |
\(\approx\) |
\(0.9511875000 + 0.1379616514i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
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
−27.11979771495714741384021089929, −26.67162154216486509655139193875, −25.37360311810173662269537282991, −24.79244505983723237569094505876, −23.07084918170088983442206389195, −22.017079217120998753449592046250, −21.14268427585445469725217207692, −20.24825757872031720797264468812, −19.75348716678115056356304622545, −18.59959712480627007587747600196, −17.4005611753790605904025887582, −16.53466122991678343364327356230, −15.67857363972117715396240763243, −14.1836685335411076328176514063, −13.48630965223452754084417417357, −11.95036189466682327186649529528, −10.83201501461709401067234671280, −10.159499050286744130126225864500, −9.10537557177132655940956419758, −8.11227919174078404866020008202, −7.21461917899420123250662731352, −5.130362112345306518366513761128, −3.70910989021222475673396303650, −2.99688392416747487210908974140, −1.20440530579781682804662361289,
1.49274826850382092551980259398, 2.48727408213290032097280105305, 4.59466836911135843608944743859, 6.13758243073874624471831903862, 6.96434289691820697156212199635, 8.08351211456533834401048178890, 8.95207606688127421093649932900, 9.74238218245469515348941674052, 11.425324621057373988116211231012, 12.36388200023622822228560023209, 13.76613242104494998494776226693, 14.80584301045380773358393540278, 15.27755800387833394504945133992, 16.78645411446527414115109771288, 17.74068759054266198617037600554, 18.54282379807014503664599819950, 19.33175010180948005349172887017, 20.17041177109696315410253511403, 21.37509697378750647443305163854, 22.81536336701759281260535889147, 23.98045220032302844519655914176, 24.54180768465872038513188581479, 25.40760747654367082043284464507, 26.12049094823259017309424831424, 27.06895589435116657665039087758