L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.5 + 0.866i)3-s + (0.766 − 0.642i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)6-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + 10-s + (−0.939 − 0.342i)11-s + (0.173 + 0.984i)12-s + (0.173 − 0.984i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)15-s + (0.173 − 0.984i)16-s + (−0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.5 + 0.866i)3-s + (0.766 − 0.642i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)6-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + 10-s + (−0.939 − 0.342i)11-s + (0.173 + 0.984i)12-s + (0.173 − 0.984i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)15-s + (0.173 − 0.984i)16-s + (−0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.0003761642187 + 0.0007952982925i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0003761642187 + 0.0007952982925i\) |
\(L(1)\) |
\(\approx\) |
\(0.3154054031 + 0.1062370111i\) |
\(L(1)\) |
\(\approx\) |
\(0.3154054031 + 0.1062370111i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 73 | \( 1 \) |
good | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.939 - 0.342i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.939 - 0.342i)T \) |
| 13 | \( 1 + (0.173 - 0.984i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.939 + 0.342i)T \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.939 + 0.342i)T \) |
| 31 | \( 1 + (0.766 + 0.642i)T \) |
| 37 | \( 1 + (-0.939 - 0.342i)T \) |
| 41 | \( 1 + (0.173 + 0.984i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.173 + 0.984i)T \) |
| 53 | \( 1 + (-0.939 + 0.342i)T \) |
| 59 | \( 1 + (0.173 - 0.984i)T \) |
| 61 | \( 1 + (0.173 + 0.984i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.939 + 0.342i)T \) |
| 79 | \( 1 + (0.173 - 0.984i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.173 - 0.984i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−31.50609331591354884854874143647, −30.48351022698545950351445950232, −29.735763687875598153516185834250, −28.622369434353501040929375822607, −27.87476453790483251237694786442, −26.356516680425618393630459480149, −25.921973503345439686488673584307, −24.09854807943550545689824276364, −23.49655581371472136808428731109, −22.15522766586045971000535812381, −20.508914017755331238317005123509, −19.36184256893714617578399268556, −18.863411049649895202749691319434, −17.56410064105683586866719926773, −16.60106090874821532995905874771, −15.45634958471339043239582741096, −13.43012849461408740185968287904, −12.30381692943762508598756647872, −11.18514900702503695802821831188, −10.31237438303233743589173329867, −8.39004411098272818686704314841, −7.358166417797208855069758069483, −6.50171810659043083810630672632, −3.95659730473704639674324233698, −2.10856525779833231432838959504,
0.00134915037009762164350461399, 3.019875924697657868091694030547, 5.02138989001083051392780279034, 6.20812263444092419308258816168, 8.0056051820264156693681495257, 8.98796722379019432113373115391, 10.29744417325347370867533898458, 11.325321784764333560376475314910, 12.52086689449267952797979858848, 14.93457232126758618414141083604, 15.82771929542678745046695249066, 16.24355710698557663543939811401, 17.7558812820232803553823482261, 18.79872742675125895608916810406, 20.06009167379724342291590174074, 21.016727408748985259820173356891, 22.557778889359616093168473701049, 23.55214146793791245975136326516, 24.78142677373471535787167462287, 26.040067216542253767669876281094, 26.9774590981860681140879374, 27.930078443627321353793614416944, 28.47058359356749118328307739594, 29.65306692387886602389086521987, 31.65364137396657137377235990277