L(s) = 1 | + (0.433 + 0.900i)2-s + (0.781 − 0.623i)3-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.900 + 0.433i)6-s + (−0.623 − 0.781i)7-s + (−0.974 − 0.222i)8-s + (0.222 − 0.974i)9-s + (−0.781 − 0.623i)10-s + (0.974 − 0.222i)11-s + i·12-s + (0.222 + 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.433 + 0.900i)15-s + (−0.222 − 0.974i)16-s + i·17-s + ⋯ |
L(s) = 1 | + (0.433 + 0.900i)2-s + (0.781 − 0.623i)3-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.900 + 0.433i)6-s + (−0.623 − 0.781i)7-s + (−0.974 − 0.222i)8-s + (0.222 − 0.974i)9-s + (−0.781 − 0.623i)10-s + (0.974 − 0.222i)11-s + i·12-s + (0.222 + 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.433 + 0.900i)15-s + (−0.222 − 0.974i)16-s + i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.448773913 + 1.014605518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.448773913 + 1.014605518i\) |
\(L(1)\) |
\(\approx\) |
\(1.261965840 + 0.5126558305i\) |
\(L(1)\) |
\(\approx\) |
\(1.261965840 + 0.5126558305i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.433 + 0.900i)T \) |
| 3 | \( 1 + (0.781 - 0.623i)T \) |
| 5 | \( 1 + (-0.900 + 0.433i)T \) |
| 7 | \( 1 + (-0.623 - 0.781i)T \) |
| 11 | \( 1 + (0.974 - 0.222i)T \) |
| 13 | \( 1 + (0.222 + 0.974i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (0.781 + 0.623i)T \) |
| 31 | \( 1 + (0.433 + 0.900i)T \) |
| 37 | \( 1 + (0.974 + 0.222i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (0.433 - 0.900i)T \) |
| 47 | \( 1 + (0.974 - 0.222i)T \) |
| 53 | \( 1 + (0.900 - 0.433i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (-0.781 + 0.623i)T \) |
| 67 | \( 1 + (-0.222 + 0.974i)T \) |
| 71 | \( 1 + (0.222 + 0.974i)T \) |
| 73 | \( 1 + (0.433 - 0.900i)T \) |
| 79 | \( 1 + (-0.974 - 0.222i)T \) |
| 83 | \( 1 + (-0.623 + 0.781i)T \) |
| 89 | \( 1 + (-0.433 - 0.900i)T \) |
| 97 | \( 1 + (-0.781 - 0.623i)T \) |
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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
−22.51425494395760614339722093381, −21.85625635111060128312801119577, −20.77792597807871413606815617, −20.17235779938880758772383983763, −19.658344918607483787829703549777, −18.94092223939519205127255592583, −18.0433815863760851818209435160, −16.63324887022198730846652076959, −15.55229520176226787010895505413, −15.326198549707112411831517609857, −14.27019441871903480443189782362, −13.37026891675324444292643086252, −12.54964098101486613715891750807, −11.74580899509708667257614807039, −10.98938079089641090744051422744, −9.72606870409610745007785075025, −9.2635012920067396303988096994, −8.48513158307856442425959458172, −7.33961004154480871553822165568, −5.81143366725373135975726848275, −4.866310330761058416301425693169, −4.00897456725948188079101406197, −3.18125738133148457974034848856, −2.46543672200743665590556603477, −0.86036845130599718024641307316,
1.13877879361464869487699720997, 2.91758935389216278240474384664, 3.86099127823308835080559950631, 4.13707489126431149495237344126, 6.04984859142776982086146352958, 6.79040459024031447497966109368, 7.31207134051242963946111470820, 8.25622782088107303115563923992, 8.983733369391908103016700080869, 10.05101830527459291489156292573, 11.56152822470227344367682908093, 12.24930701796219880328619297522, 13.17332555148179481130583397931, 14.0619970908434271952413030201, 14.46872124859524782626989553190, 15.38219483301223334104367950793, 16.29095034711495428471884538167, 16.94758345942228433014197751976, 18.068461920414479635085237221, 18.95400180906099178053058431637, 19.530743091793804248883365083982, 20.34549115254026173157835361108, 21.482751032356122621529071766, 22.38870213997677690513287633784, 23.23748316899057299792340464136