L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − i·7-s − i·8-s − 9-s − 11-s + i·12-s + i·13-s + 14-s + 16-s + i·17-s − i·18-s + 19-s + ⋯ |
L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − i·7-s − i·8-s − 9-s − 11-s + i·12-s + i·13-s + 14-s + 16-s + i·17-s − i·18-s + 19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1327780773 + 0.4673983536i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1327780773 + 0.4673983536i\) |
\(L(1)\) |
\(\approx\) |
\(0.7181825133 + 0.1695398934i\) |
\(L(1)\) |
\(\approx\) |
\(0.7181825133 + 0.1695398934i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 - iT \) |
| 31 | \( 1 \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 - T \) |
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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
−27.7143104810890111499065142720, −26.89983841097122089759422945065, −25.96604711079314446328363090984, −24.73570012047173502388604376112, −23.1009350254349437185084530877, −22.402417711784745664339052279718, −21.550514236807187613059373138, −20.65291023044658844305401056725, −19.98625069413349630555384885337, −18.48961833304613698058920864476, −17.91915421737249826412408787354, −16.36346267144356927167549561217, −15.37751002639882368492909100118, −14.339207226433258900638505618605, −13.04222034105384495110720690682, −11.94710280664158824333703641040, −10.936540857919177431492193759118, −9.98604804971970369560354125999, −9.064363042206127597179459902515, −8.0134591485702585425419823569, −5.53981590717756164550826273906, −4.95315792226224274779853956672, −3.312335818245313523698763724857, −2.49213864964311873246965093286, −0.193091479238273279490287591756,
1.42149042415977711792007498901, 3.530957214897630374629223095770, 5.07311868305124461884397163197, 6.30536869289966290129006390465, 7.33976780154230557799289571923, 7.989110608410336536275705163857, 9.349716858651454772234956395, 10.77538338935851246254703179215, 12.29259016626432357643106475635, 13.49241658390150292192675219986, 13.87946331213049852836291351652, 15.19970240378666719054713922017, 16.50373105614836051033003886817, 17.26363363655423010310464803561, 18.253622117889194392236657977316, 19.09903746857010971055706997396, 20.24274633732597637545827839535, 21.714735993141600231585492641543, 22.976200865565071361039252869663, 23.87187882769789618288287073048, 24.061460265257668265555824120983, 25.587503297953535432848239844761, 26.10641678245437450658895871999, 27.08787866597317488041874979587, 28.50848115353290942385531932976