L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)5-s + (0.900 − 0.433i)8-s + (−0.900 − 0.433i)10-s + (−0.623 − 0.781i)11-s + (0.623 + 0.781i)13-s + (−0.900 − 0.433i)16-s + (0.222 + 0.974i)17-s + 19-s + (0.222 + 0.974i)20-s + (−0.222 + 0.974i)22-s + (0.222 − 0.974i)23-s + (0.623 − 0.781i)25-s + (0.222 − 0.974i)26-s + ⋯ |
L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)5-s + (0.900 − 0.433i)8-s + (−0.900 − 0.433i)10-s + (−0.623 − 0.781i)11-s + (0.623 + 0.781i)13-s + (−0.900 − 0.433i)16-s + (0.222 + 0.974i)17-s + 19-s + (0.222 + 0.974i)20-s + (−0.222 + 0.974i)22-s + (0.222 − 0.974i)23-s + (0.623 − 0.781i)25-s + (0.222 − 0.974i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.219652185 - 0.9102500826i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.219652185 - 0.9102500826i\) |
\(L(1)\) |
\(\approx\) |
\(0.9038644626 - 0.3925975818i\) |
\(L(1)\) |
\(\approx\) |
\(0.9038644626 - 0.3925975818i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 5 | \( 1 + (0.900 - 0.433i)T \) |
| 11 | \( 1 + (-0.623 - 0.781i)T \) |
| 13 | \( 1 + (0.623 + 0.781i)T \) |
| 17 | \( 1 + (0.222 + 0.974i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.222 - 0.974i)T \) |
| 29 | \( 1 + (0.222 + 0.974i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.222 - 0.974i)T \) |
| 41 | \( 1 + (0.900 - 0.433i)T \) |
| 43 | \( 1 + (-0.900 - 0.433i)T \) |
| 47 | \( 1 + (-0.623 - 0.781i)T \) |
| 53 | \( 1 + (0.222 - 0.974i)T \) |
| 59 | \( 1 + (0.900 + 0.433i)T \) |
| 61 | \( 1 + (-0.222 - 0.974i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.623 - 0.781i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.623 + 0.781i)T \) |
| 89 | \( 1 + (-0.623 + 0.781i)T \) |
| 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
−28.08262830171997910932535166892, −26.965381859992417394834327100828, −26.04098101334388010046298317094, −25.30015974884969979644418750326, −24.58624342121550399539338890581, −23.1341974087899829646055250545, −22.61652859475812803142486496603, −21.08222135475075872526476027363, −20.13348558674777273346941609063, −18.73397948902628187128308900482, −17.96904298287511344102490385920, −17.34434919820037332918018487846, −15.96428179469367293256382865720, −15.19366740791898415261468496937, −13.97346434640634940593673981266, −13.224341513874787975185253803482, −11.37948008693342891080458696476, −10.07839500990124649819281957184, −9.58112279470382385285403497269, −8.06800216372280144212055681177, −7.07987991371198341739698932099, −5.886419805729266949931442783881, −4.96465728301693316344549511373, −2.7708118428619932208098835354, −1.16479338495475741322263601014,
0.89643219470382860681351894080, 2.16489111688356202898283693798, 3.56638157548100251959127937470, 5.12201320137962859839804210789, 6.55768573587974566985587625923, 8.20938628091208579115107348071, 8.9900828533654905696972070552, 10.132837149668176424281975970, 11.01292655593038245107100427161, 12.30428074490307810147836921375, 13.28084542624787559603351975471, 14.13378347233710220854755583403, 16.088876275253987799647210404670, 16.77190343722071638499765031666, 17.92327486365335700641082375988, 18.661285705480357297826646897160, 19.77299611160369968380293745012, 20.999377481780721684840440926665, 21.3436567586817284799520328087, 22.47000292279971868012153309259, 23.9157604537244679613663848173, 24.97902124534904725050931857811, 26.10657631168013350997116677049, 26.62649502499334326934005948571, 28.08506031896688452015546448819