L(s) = 1 | − 2-s + 4-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 − 0.866i)10-s − 11-s + (0.5 + 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + ⋯ |
L(s) = 1 | − 2-s + 4-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 − 0.866i)10-s − 11-s + (0.5 + 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1951861729 - 0.3779022266i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1951861729 - 0.3779022266i\) |
\(L(1)\) |
\(\approx\) |
\(0.5783337552 - 0.05433636154i\) |
\(L(1)\) |
\(\approx\) |
\(0.5783337552 - 0.05433636154i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 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
−28.89609637118756700838598818409, −28.36550625815816314227632340061, −27.52386142920867788290713595015, −26.00319067717235564540562232707, −25.560126831159240710123520519884, −24.467192967164144538749441616982, −23.56599517799745689463056478720, −21.68595969391509867749345457165, −21.09026481377946409482969925509, −19.88286851121048718592077068264, −18.93249462280383310353233841939, −17.90209423674845748875851626888, −16.93360078527273806124489540182, −15.93621166226642286485914306754, −15.064513094804255953324045223930, −13.14148430931116422600554494968, −12.35451469077695574494656830971, −10.94391059481437796553337637041, −9.70910105397165985582725129654, −8.90167577036602780919817546741, −7.85591733031606308661372220461, −6.264366392263211243214864230240, −5.23056127598410788579730485809, −2.93944476720687290781792833959, −1.579024919052309978694223653101,
0.23154837220032149671351451455, 2.15900658889593221789719353227, 3.42445614460836614053338370217, 5.711849489236199189170860036852, 6.950459175789567384098542296672, 7.75089295481548137946320386245, 9.37619505638000013360358024556, 10.35262501873311727525770432879, 10.964367022897621166553647065384, 12.60317557594386651748959395668, 13.94438041016320977568287072937, 15.12953763745539797974716221746, 16.35683318321468073072354614178, 17.19824289915600597854853840254, 18.45652693304130748221808095908, 18.91777847856022021657202041046, 20.371446160245824240393372015407, 21.07667947997709611878553765606, 22.54894888024505310548896530741, 23.53325422342196508977843000526, 24.91170130524122668603411507515, 25.895981811338542000853208174496, 26.466695573065503402642625909869, 27.378760057914226441737270387012, 28.731586030574547342421447337586