L(s) = 1 | + (0.195 − 0.980i)5-s + (−0.382 + 0.923i)7-s + (0.831 − 0.555i)11-s + (0.195 + 0.980i)13-s + (−0.707 + 0.707i)17-s + (0.980 − 0.195i)19-s + (0.923 − 0.382i)23-s + (−0.923 − 0.382i)25-s + (0.831 + 0.555i)29-s − i·31-s + (0.831 + 0.555i)35-s + (0.980 + 0.195i)37-s + (0.923 − 0.382i)41-s + (−0.555 − 0.831i)43-s + (−0.707 + 0.707i)47-s + ⋯ |
L(s) = 1 | + (0.195 − 0.980i)5-s + (−0.382 + 0.923i)7-s + (0.831 − 0.555i)11-s + (0.195 + 0.980i)13-s + (−0.707 + 0.707i)17-s + (0.980 − 0.195i)19-s + (0.923 − 0.382i)23-s + (−0.923 − 0.382i)25-s + (0.831 + 0.555i)29-s − i·31-s + (0.831 + 0.555i)35-s + (0.980 + 0.195i)37-s + (0.923 − 0.382i)41-s + (−0.555 − 0.831i)43-s + (−0.707 + 0.707i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.373067694 - 0.1012835579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.373067694 - 0.1012835579i\) |
\(L(1)\) |
\(\approx\) |
\(1.137818143 - 0.05991848213i\) |
\(L(1)\) |
\(\approx\) |
\(1.137818143 - 0.05991848213i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.195 - 0.980i)T \) |
| 7 | \( 1 + (-0.382 + 0.923i)T \) |
| 11 | \( 1 + (0.831 - 0.555i)T \) |
| 13 | \( 1 + (0.195 + 0.980i)T \) |
| 17 | \( 1 + (-0.707 + 0.707i)T \) |
| 19 | \( 1 + (0.980 - 0.195i)T \) |
| 23 | \( 1 + (0.923 - 0.382i)T \) |
| 29 | \( 1 + (0.831 + 0.555i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (0.980 + 0.195i)T \) |
| 41 | \( 1 + (0.923 - 0.382i)T \) |
| 43 | \( 1 + (-0.555 - 0.831i)T \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
| 53 | \( 1 + (0.831 - 0.555i)T \) |
| 59 | \( 1 + (0.195 - 0.980i)T \) |
| 61 | \( 1 + (-0.555 + 0.831i)T \) |
| 67 | \( 1 + (0.555 - 0.831i)T \) |
| 71 | \( 1 + (-0.382 + 0.923i)T \) |
| 73 | \( 1 + (0.382 + 0.923i)T \) |
| 79 | \( 1 + (0.707 + 0.707i)T \) |
| 83 | \( 1 + (0.980 - 0.195i)T \) |
| 89 | \( 1 + (-0.923 - 0.382i)T \) |
| 97 | \( 1 + iT \) |
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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
−24.88436810585693645457755915130, −23.2569391492877284115149951994, −22.875419453494148562649466650704, −22.162530017441445544266829702311, −21.050057760543942405039711162379, −19.929691644945282340141705703366, −19.5589159552040981452737835250, −18.118481078409650168826740359, −17.73915747345491373823179428084, −16.63576692948564565464377058418, −15.607177047033658653507764267547, −14.71756565970661498620836488639, −13.83776616762712849218274831334, −13.10991178536601149559368731584, −11.81130141443916253247495032205, −10.91387387935489415755083396735, −10.05968197832216326654337604804, −9.28828109782356302239182169436, −7.74107310355707348023760505334, −6.99953942266537163500903509583, −6.2003648892996284692698415681, −4.8048597693073956252580932573, −3.58207052879051283671268256279, −2.74746588197727021343929561460, −1.11925169234553115365874977501,
1.1333852721953821429227748183, 2.37763610552161543577585115157, 3.786296553199042309098695952395, 4.85834349178004628502626639877, 5.92354327662774953685433122155, 6.73454478668644238518677633115, 8.31892504099315228363222153367, 9.02353305499266508722914636320, 9.60769764097571304457020836288, 11.18941476426804203293997491747, 11.93709941647511112406117716860, 12.84017137590938189040789288545, 13.67658858311047333395131689915, 14.73384553106353781515486177746, 15.829870461189859813630047721743, 16.50219291717802495125429227680, 17.32470489197903206610323871886, 18.42637099090648234531087341391, 19.320748066467392654815535580709, 20.04365031296908860036260756786, 21.182586122999913607819590122341, 21.774341399681644534343497529998, 22.61290079934122086143391881882, 23.92066223522327383748003669037, 24.47357771963549926571374334964