L(s) = 1 | + i·7-s − 11-s − i·13-s + i·17-s − 19-s + 29-s − 31-s − i·37-s − 41-s − i·43-s + i·47-s − 49-s − i·53-s − 59-s − 61-s + ⋯ |
L(s) = 1 | + i·7-s − 11-s − i·13-s + i·17-s − 19-s + 29-s − 31-s − i·37-s − 41-s − i·43-s + i·47-s − 49-s − i·53-s − 59-s − 61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03830942612 - 0.1348548122i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03830942612 - 0.1348548122i\) |
\(L(1)\) |
\(\approx\) |
\(0.7774104452 + 0.04766107499i\) |
\(L(1)\) |
\(\approx\) |
\(0.7774104452 + 0.04766107499i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 + iT \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 \) |
| 89 | \( 1 \) |
| 97 | \( 1 \) |
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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
−21.224430573337463098370045042859, −20.29733486162613454231012028605, −19.80727924228498933544690457332, −18.71675935230276150404246979395, −18.30392359723409249586705022388, −17.20619497834119359603298252303, −16.658922077576046667043648264876, −15.92672841840245060425755634525, −15.081622016732097829893009556556, −14.096114113827146544823906101381, −13.60663950939625713278389363213, −12.81991828293232312256076882729, −11.86962252508924813385728195154, −11.02027434352566311931554905358, −10.36417544731906270668041037469, −9.580867848056517336607206948261, −8.62054769948759405258208312736, −7.75071288149972188552777417448, −6.989535753402340651471841527788, −6.278544611587921432574273172634, −4.9898980045449165042187811913, −4.463298218534429423599834624482, −3.40283521274873333351786387928, −2.426777381295018796154762611130, −1.33601727162627164439862532628,
0.051578988228225601488025118008, 1.72848120574002898056173754312, 2.5729830509538867328661187038, 3.42226315485556351308137053767, 4.59775599092969578760637210075, 5.525026098804263756121073228254, 6.03274791258236621019932896679, 7.15991022801519378061559110904, 8.24122467951629954526740724211, 8.54075551626280376862520854304, 9.685029452868573578915103479762, 10.550667747473064253013173013653, 11.07184920869802278652471728960, 12.46499454858465312006155555474, 12.55969581401634599879190382245, 13.52180882153785707049459784947, 14.62238136196256665703681417144, 15.2822323608318184631329720406, 15.75150238101366737400486316193, 16.74799238953460094468991785803, 17.66326753089276495588779615815, 18.204923643085015242396659651358, 19.011384811773458261458762499519, 19.67459363036656204933667802608, 20.61286039109536322386410294834