L(s) = 1 | + (−0.272 − 0.962i)2-s + (0.978 − 0.207i)3-s + (−0.851 + 0.524i)4-s + (−0.861 + 0.508i)5-s + (−0.466 − 0.884i)6-s + (0.736 + 0.676i)8-s + (0.913 − 0.406i)9-s + (0.723 + 0.690i)10-s + (−0.723 + 0.690i)12-s + (0.610 + 0.791i)13-s + (−0.736 + 0.676i)15-s + (0.449 − 0.893i)16-s + (−0.905 + 0.424i)17-s + (−0.640 − 0.768i)18-s + (0.123 − 0.992i)19-s + (0.466 − 0.884i)20-s + ⋯ |
L(s) = 1 | + (−0.272 − 0.962i)2-s + (0.978 − 0.207i)3-s + (−0.851 + 0.524i)4-s + (−0.861 + 0.508i)5-s + (−0.466 − 0.884i)6-s + (0.736 + 0.676i)8-s + (0.913 − 0.406i)9-s + (0.723 + 0.690i)10-s + (−0.723 + 0.690i)12-s + (0.610 + 0.791i)13-s + (−0.736 + 0.676i)15-s + (0.449 − 0.893i)16-s + (−0.905 + 0.424i)17-s + (−0.640 − 0.768i)18-s + (0.123 − 0.992i)19-s + (0.466 − 0.884i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.281470453 + 0.009679788428i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.281470453 + 0.009679788428i\) |
\(L(1)\) |
\(\approx\) |
\(1.000647573 - 0.2543476693i\) |
\(L(1)\) |
\(\approx\) |
\(1.000647573 - 0.2543476693i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.272 - 0.962i)T \) |
| 3 | \( 1 + (0.978 - 0.207i)T \) |
| 5 | \( 1 + (-0.861 + 0.508i)T \) |
| 13 | \( 1 + (0.610 + 0.791i)T \) |
| 17 | \( 1 + (-0.905 + 0.424i)T \) |
| 19 | \( 1 + (0.123 - 0.992i)T \) |
| 23 | \( 1 + (-0.888 + 0.458i)T \) |
| 29 | \( 1 + (-0.516 + 0.856i)T \) |
| 31 | \( 1 + (0.179 + 0.983i)T \) |
| 37 | \( 1 + (0.997 - 0.0760i)T \) |
| 41 | \( 1 + (0.897 + 0.441i)T \) |
| 43 | \( 1 + (-0.415 + 0.909i)T \) |
| 47 | \( 1 + (-0.640 + 0.768i)T \) |
| 53 | \( 1 + (0.449 + 0.893i)T \) |
| 59 | \( 1 + (0.830 + 0.556i)T \) |
| 61 | \( 1 + (-0.969 + 0.244i)T \) |
| 67 | \( 1 + (0.928 - 0.371i)T \) |
| 71 | \( 1 + (0.974 + 0.226i)T \) |
| 73 | \( 1 + (0.964 - 0.263i)T \) |
| 79 | \( 1 + (-0.749 - 0.662i)T \) |
| 83 | \( 1 + (-0.362 + 0.931i)T \) |
| 89 | \( 1 + (0.327 + 0.945i)T \) |
| 97 | \( 1 + (0.870 - 0.491i)T \) |
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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
−22.43343347801559996507903580002, −21.1560461985861920011288387360, −20.22894990957174990742178737325, −19.843208979250234861831457363013, −18.73729755400421341963655927943, −18.33296237396175081055581794437, −17.106785112617227537514673601704, −16.217816305365017175737955843152, −15.67141527588758642794815967232, −15.07695414630170125910181346193, −14.24126138569267109018693659775, −13.32244629429130678895355772498, −12.75999498844180921728897387278, −11.4387906055859880195311759520, −10.28466413455386713571284887909, −9.50771196385615666221914667596, −8.57475946916984020622308483160, −8.066881346847103468703451619441, −7.45061327556096719156092015636, −6.292542410131262949968541079958, −5.19532340181613918261789778759, −4.15999410258952423346641180216, −3.6634740292070939464195573191, −2.09854033363075700098370325049, −0.63989759716959922222026318664,
1.21253930766514708724360846452, 2.28710257411014593008940607188, 3.13396366426880318109990998962, 3.967408437184470279047711766, 4.610940445153422116069394721542, 6.476644577528690003485970572457, 7.391008138590700964281961109577, 8.2129811893415275152238459166, 8.926949086205094610544739648043, 9.66587445407336242710284992669, 10.83451950804992639784667362542, 11.351044208063324640291996752790, 12.34253646576450775612179482094, 13.14697403136757984950281078765, 13.92368545239024537587856983800, 14.66514900998012224968352071583, 15.62608804825923751551198959275, 16.40829712128706792622431288242, 17.88180370602543000077220638120, 18.26634966157291291492143792624, 19.16896231105586844739352157253, 19.86315002641378179201685199077, 20.06824740737868478353717893346, 21.3697226896787384807245349617, 21.73276326188559980350596729414