L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.809 − 0.587i)6-s − i·7-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.309 + 0.951i)11-s + (0.951 + 0.309i)12-s + (−0.951 − 0.309i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.587 + 0.809i)17-s − i·18-s + (0.809 − 0.587i)21-s + (−0.587 − 0.809i)22-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.809 − 0.587i)6-s − i·7-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.309 + 0.951i)11-s + (0.951 + 0.309i)12-s + (−0.951 − 0.309i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.587 + 0.809i)17-s − i·18-s + (0.809 − 0.587i)21-s + (−0.587 − 0.809i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2779135500 + 0.7019295020i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2779135500 + 0.7019295020i\) |
\(L(1)\) |
\(\approx\) |
\(0.6517394925 + 0.3622172112i\) |
\(L(1)\) |
\(\approx\) |
\(0.6517394925 + 0.3622172112i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 + (0.587 + 0.809i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (0.309 + 0.951i)T \) |
| 13 | \( 1 + (-0.951 - 0.309i)T \) |
| 17 | \( 1 + (-0.587 + 0.809i)T \) |
| 23 | \( 1 + (-0.951 + 0.309i)T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.809 + 0.587i)T \) |
| 37 | \( 1 + (0.951 + 0.309i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (0.587 + 0.809i)T \) |
| 53 | \( 1 + (0.587 + 0.809i)T \) |
| 59 | \( 1 + (0.309 - 0.951i)T \) |
| 61 | \( 1 + (0.309 + 0.951i)T \) |
| 67 | \( 1 + (0.587 - 0.809i)T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.951 + 0.309i)T \) |
| 79 | \( 1 + (-0.809 + 0.587i)T \) |
| 83 | \( 1 + (0.587 - 0.809i)T \) |
| 89 | \( 1 + (0.309 + 0.951i)T \) |
| 97 | \( 1 + (-0.587 - 0.809i)T \) |
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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
−24.04036003370285709765712294619, −22.382051796472326072359319771877, −21.67634033984477836622391230194, −20.669199649735922764706781561813, −19.886481802360747542146165814000, −19.02500096339739228093603531456, −18.63158245112451719187721813403, −17.752328763451807347614873190, −16.86139429744979240287774052860, −15.80620078692389677175979106763, −14.94066224697025708753205093328, −13.89207778829754007878140948602, −12.86413873609224377796304457389, −11.83667333467019552689733120700, −11.54801146228659463158608898873, −9.97146205133424205187256306123, −9.10385760666157524462794938229, −8.51314556732899202121930821059, −7.578543008813114522737416430625, −6.64692248948318422602313929253, −5.71428048602336935718197664821, −3.828860499111560694802664579676, −2.55431168208992679430489969813, −2.120650402133815512111415691390, −0.504691725831594059558312230307,
1.56212460132342873123090894021, 2.69504906630444064563751633543, 4.05352751055973082535976527827, 4.967119416410143960171808012737, 6.38290549336021527585845142284, 7.470597906808018460809330733765, 8.06571684253400122359232600286, 9.25331266860814512806292655182, 9.95693204247832074669997415902, 10.52263730861978265897797387973, 11.54168274033046468435691931223, 12.87914242880683382665942075257, 14.16409529154992125915184162806, 14.82104470917191577509431122364, 15.551786018013792551437028064342, 16.55209020369281409278594187128, 17.19838890694553526722602816120, 17.93403307921264799365636792384, 19.30947715275087941467263927360, 20.01953482680709813783318079943, 20.247899408098180787100146613694, 21.430050247783912719230113468582, 22.39100115545489649874036540069, 23.43543522901276312327733708538, 24.38449929436642311178961641789