L(s) = 1 | − 2-s + (−0.5 + 0.866i)3-s + 4-s + (0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s − 7-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + 14-s + (0.5 + 0.866i)15-s + 16-s + 17-s + (0.5 + 0.866i)18-s + ⋯ |
L(s) = 1 | − 2-s + (−0.5 + 0.866i)3-s + 4-s + (0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s − 7-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + 14-s + (0.5 + 0.866i)15-s + 16-s + 17-s + (0.5 + 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7168958172 - 0.08808863440i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7168958172 - 0.08808863440i\) |
\(L(1)\) |
\(\approx\) |
\(0.6089104368 + 0.04436462080i\) |
\(L(1)\) |
\(\approx\) |
\(0.6089104368 + 0.04436462080i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 103 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 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
−20.761083995108105809045533797993, −19.810862383242555619525675839040, −19.07914458851228472882588925660, −18.65093621850171227241598072127, −18.15529728285056925673996992828, −17.1674533111801426324179851738, −16.56278947051805036830978289061, −16.06386418050451838262419271317, −14.75780857282433027986294670133, −14.0636637093923807760210631470, −13.20732702342292983307389054904, −12.11792487051978997141440372254, −11.74857228696620654509955871342, −10.55774486735713882191810632073, −10.25038328093796234794928889863, −9.268290909155789005955913047, −8.28434159998722768639825375200, −7.47853365887140027286381746460, −6.64052938836412431223339507727, −6.17315941633253056526663203040, −5.54090415914318122067969788734, −3.471791260071534454910706524898, −2.86206229574784546368875250564, −1.81282000060797499252803283238, −0.83312219634445668910739595527,
0.57697787425072492324957187602, 1.662701802067004846717286954602, 2.94355299268717800658726070041, 3.856671236460026985746114147897, 5.03257885532991675704794423431, 5.80763685723596496680850390675, 6.576689826561771357572691189247, 7.50642752726914738940563272920, 8.70773431560649543832308852552, 9.42498753760389778326602306860, 9.76350848812313697893142660130, 10.41612503041661025819108384391, 11.58570005331762743420852826068, 12.16728491800852078250596886006, 12.8990512125312348264290395083, 14.14244965361415216792099077149, 15.15116058970473596058253590907, 15.91232271801741434705141822931, 16.3694418127921489813455454569, 17.09534339110711135537997253420, 17.62012700296174896776782415793, 18.39599420606576236426419124986, 19.63455126812881469579047622602, 20.040498041489785389305447401027, 20.6846572716025323284570878424