L(s) = 1 | + (−0.669 − 0.743i)2-s + (0.104 − 0.994i)3-s + (−0.104 + 0.994i)4-s + (−0.809 + 0.587i)6-s + (0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.978 + 0.207i)11-s + (0.978 + 0.207i)12-s + (−0.309 − 0.951i)13-s + (−0.978 − 0.207i)16-s + (−0.913 − 0.406i)17-s + (0.5 + 0.866i)18-s + (−0.104 − 0.994i)19-s + (0.809 + 0.587i)22-s + (−0.669 − 0.743i)23-s + (−0.5 − 0.866i)24-s + ⋯ |
L(s) = 1 | + (−0.669 − 0.743i)2-s + (0.104 − 0.994i)3-s + (−0.104 + 0.994i)4-s + (−0.809 + 0.587i)6-s + (0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.978 + 0.207i)11-s + (0.978 + 0.207i)12-s + (−0.309 − 0.951i)13-s + (−0.978 − 0.207i)16-s + (−0.913 − 0.406i)17-s + (0.5 + 0.866i)18-s + (−0.104 − 0.994i)19-s + (0.809 + 0.587i)22-s + (−0.669 − 0.743i)23-s + (−0.5 − 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.05912464177 - 0.4669993766i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.05912464177 - 0.4669993766i\) |
\(L(1)\) |
\(\approx\) |
\(0.4239781971 - 0.4330713274i\) |
\(L(1)\) |
\(\approx\) |
\(0.4239781971 - 0.4330713274i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.669 - 0.743i)T \) |
| 3 | \( 1 + (0.104 - 0.994i)T \) |
| 11 | \( 1 + (-0.978 + 0.207i)T \) |
| 13 | \( 1 + (-0.309 - 0.951i)T \) |
| 17 | \( 1 + (-0.913 - 0.406i)T \) |
| 19 | \( 1 + (-0.104 - 0.994i)T \) |
| 23 | \( 1 + (-0.669 - 0.743i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.913 + 0.406i)T \) |
| 37 | \( 1 + (0.978 + 0.207i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.913 + 0.406i)T \) |
| 53 | \( 1 + (0.104 - 0.994i)T \) |
| 59 | \( 1 + (0.669 - 0.743i)T \) |
| 61 | \( 1 + (0.669 + 0.743i)T \) |
| 67 | \( 1 + (-0.913 - 0.406i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.978 - 0.207i)T \) |
| 79 | \( 1 + (0.913 - 0.406i)T \) |
| 83 | \( 1 + (0.809 - 0.587i)T \) |
| 89 | \( 1 + (0.669 + 0.743i)T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.814020231648725958183380229808, −26.69107259828929651458059739935, −26.330052527054533415833874882609, −25.36792239066265822423800158627, −24.17800827675847013278935408284, −23.33883976778571039817444386417, −22.20457175417409079664020628673, −21.176589511740499925246276044821, −20.12739995145778706230380122597, −19.144363374741623743293802795841, −18.06487593592610479393568405882, −16.95054513113265396292027048684, −16.201265805324462030131259601034, −15.36118184151160105243611567129, −14.470242463255042214426023775478, −13.434663850861266095935557788836, −11.543122881821613801618854954273, −10.52177576996073178002524594982, −9.678392874648017623821835006616, −8.67760237056671287845793710893, −7.70235422699559014109851452642, −6.222394618992766235139098394748, −5.18353201464658707534360567208, −4.02161672439618875848739685891, −2.15891177536172109607440030960,
0.43872940117787875001333213541, 2.1703178679136334918387635186, 2.96806069751656257925410395166, 4.8220436451510530022924536420, 6.53797378841781765480681660417, 7.699410557306215871181572250855, 8.400740603107979674424338462850, 9.70348564889078920397797220360, 10.883193677100364857046953324353, 11.83497876170215119967850263633, 12.98613893936222013047560870162, 13.40034230342831368233826992767, 15.069946992869461261958582310584, 16.38262525347989835079526354062, 17.82022633283989060197071683215, 17.93600696237593234414288668258, 19.16927138609531215653255714373, 20.01129426300333793673451998247, 20.7223504623699400916238720618, 22.089615587180995631498393109130, 22.9710167990890110976311930986, 24.21762305220172439635651769751, 25.118950358764121276460329304519, 26.09728311060925853129067454080, 26.83424699399941610471310339103