L(s) = 1 | + (−0.642 + 0.766i)5-s + (0.642 + 0.766i)11-s + (−0.342 − 0.939i)13-s + 17-s − i·19-s + (0.939 − 0.342i)23-s + (−0.173 − 0.984i)25-s + (0.342 − 0.939i)29-s + (0.173 − 0.984i)31-s + (0.866 − 0.5i)37-s + (0.939 − 0.342i)41-s + (0.984 − 0.173i)43-s + (−0.173 − 0.984i)47-s + (−0.866 + 0.5i)53-s − 55-s + ⋯ |
L(s) = 1 | + (−0.642 + 0.766i)5-s + (0.642 + 0.766i)11-s + (−0.342 − 0.939i)13-s + 17-s − i·19-s + (0.939 − 0.342i)23-s + (−0.173 − 0.984i)25-s + (0.342 − 0.939i)29-s + (0.173 − 0.984i)31-s + (0.866 − 0.5i)37-s + (0.939 − 0.342i)41-s + (0.984 − 0.173i)43-s + (−0.173 − 0.984i)47-s + (−0.866 + 0.5i)53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7983102236 - 1.110374579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7983102236 - 1.110374579i\) |
\(L(1)\) |
\(\approx\) |
\(0.9874222667 - 0.04170219036i\) |
\(L(1)\) |
\(\approx\) |
\(0.9874222667 - 0.04170219036i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.642 + 0.766i)T \) |
| 11 | \( 1 + (0.642 + 0.766i)T \) |
| 13 | \( 1 + (-0.342 - 0.939i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + (0.939 - 0.342i)T \) |
| 29 | \( 1 + (0.342 - 0.939i)T \) |
| 31 | \( 1 + (0.173 - 0.984i)T \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 + (0.984 - 0.173i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 + (0.342 + 0.939i)T \) |
| 61 | \( 1 + (-0.984 + 0.173i)T \) |
| 67 | \( 1 + (-0.642 + 0.766i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (0.342 - 0.939i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.173 - 0.984i)T \) |
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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
−19.16751117679815307885572360282, −18.64053058239629702848540024018, −17.520583124347928339864033274429, −16.83359287421484373158719172476, −16.312319588015226931539074165764, −15.88458836943280493567583660254, −14.61483492067524762318912857383, −14.40783609378013021194611149848, −13.46597255904291135313101683243, −12.50144750779625204699631740309, −12.16725425225019482531894993963, −11.35322120548578013613843824067, −10.74293609373467998049824313638, −9.52873484155107056947472829606, −9.18622098063658470393630032267, −8.2659631104283027784115202108, −7.72057667405931876625434575760, −6.806189170127625347008513981405, −5.99162132042733569090751943156, −5.12403516962343539877543903198, −4.41458842358853175917746059533, −3.604070982428464312522380849812, −2.92246060862014456833249930158, −1.41630935904613582598353118446, −1.08995755131830857364043445526,
0.25512628893855260664743856542, 1.07438729425931220430911941393, 2.49230879403589316701692735703, 2.897917227527706043519909394613, 3.988949090103789384102865052145, 4.54250094501690294317544287653, 5.61385623341790089433307995283, 6.37194559359841238933097086457, 7.39796145195592784344238354901, 7.51899321899808537291474833478, 8.57976527555846648360509337451, 9.50065184488699099866481182893, 10.12676266536289751078980142289, 10.91431579239031193369364497875, 11.55571854715886102941362482558, 12.29503025538969895896367498024, 12.90390799086691990017665275977, 13.86387321212544792478236934564, 14.67720808284370441680183788990, 15.102825613087592472002290827292, 15.658492988733349747332034518050, 16.640302712151866081612167812561, 17.36244197868951857849037550297, 17.92431977610350784595876272185, 18.75384283651026249810955794389