L(s) = 1 | + (0.707 + 0.707i)2-s + (0.382 + 0.923i)3-s + i·4-s + (−0.382 + 0.923i)6-s + (0.923 + 0.382i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.923 + 0.382i)12-s + i·13-s + (0.382 + 0.923i)14-s − 16-s − 18-s + (−0.707 − 0.707i)19-s + i·21-s + (0.923 − 0.382i)22-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (0.382 + 0.923i)3-s + i·4-s + (−0.382 + 0.923i)6-s + (0.923 + 0.382i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.923 + 0.382i)12-s + i·13-s + (0.382 + 0.923i)14-s − 16-s − 18-s + (−0.707 − 0.707i)19-s + i·21-s + (0.923 − 0.382i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8458585200 + 2.636701922i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8458585200 + 2.636701922i\) |
\(L(1)\) |
\(\approx\) |
\(1.222907019 + 1.329797016i\) |
\(L(1)\) |
\(\approx\) |
\(1.222907019 + 1.329797016i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.382 + 0.923i)T \) |
| 7 | \( 1 + (0.923 + 0.382i)T \) |
| 11 | \( 1 + (0.382 - 0.923i)T \) |
| 13 | \( 1 + iT \) |
| 19 | \( 1 + (-0.707 - 0.707i)T \) |
| 23 | \( 1 + (0.382 - 0.923i)T \) |
| 29 | \( 1 + (0.923 - 0.382i)T \) |
| 31 | \( 1 + (0.382 + 0.923i)T \) |
| 37 | \( 1 + (0.382 + 0.923i)T \) |
| 41 | \( 1 + (-0.923 - 0.382i)T \) |
| 43 | \( 1 + (0.707 - 0.707i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (0.707 + 0.707i)T \) |
| 59 | \( 1 + (0.707 - 0.707i)T \) |
| 61 | \( 1 + (0.923 + 0.382i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.382 - 0.923i)T \) |
| 73 | \( 1 + (0.923 - 0.382i)T \) |
| 79 | \( 1 + (0.382 - 0.923i)T \) |
| 83 | \( 1 + (-0.707 - 0.707i)T \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + (-0.923 + 0.382i)T \) |
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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
−30.13586291538842794892539013401, −29.49359577803000134022033645502, −28.13105688134720187889145980003, −27.20204209505094747087811499277, −25.422125393934195118527596556462, −24.62157050376255418914359441385, −23.48176001334890680968250617360, −22.841292922407903876603111457066, −21.248068953436316514788339921021, −20.30576023351376405946492529539, −19.56084422572777293208064890869, −18.25258635735802592412000074362, −17.38347807919265733124080167183, −15.14120479264674469119780416893, −14.42801022126311477536727140041, −13.280360910319779752789449224524, −12.345452993171040257348474892414, −11.28178553358783174371882528852, −9.91199781708802367215867869735, −8.26735675193185876685442882342, −6.94360471319390820094845839243, −5.44988301727569002248497975148, −3.92660736935808660828905231124, −2.3155096388601637717665413763, −1.097666725800881156866635167080,
2.623723210352441436055995810156, 4.15083963487392808727723600438, 5.06931258311018511964752096431, 6.52154182895959504493356685333, 8.299978202815468157583276487625, 8.94519356569426358490929517547, 10.90948614892395900405500010462, 11.93183215971682955283218943237, 13.71839713925782094831075561698, 14.44218737640870354698460080078, 15.4177344656181425648608459947, 16.47264075571347588190011853712, 17.40770679834893736125005951426, 19.07605639923805084328362238027, 20.66001000251621456266737823738, 21.48683995432357899273698262688, 22.10155760150345028650620336957, 23.571811578818818661830651326327, 24.535499292441368222185851331931, 25.542377493003169738142178093023, 26.68355852917175758568718516130, 27.31386634679520174093114500130, 28.74272319736784949352742046221, 30.401737231785802927633855354113, 31.08845370926954442904328683479