L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + 8-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s − 14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 22-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)26-s + (0.5 + 0.866i)28-s + (0.5 − 0.866i)29-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + 8-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s − 14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 22-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)26-s + (0.5 + 0.866i)28-s + (0.5 − 0.866i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2639307870 - 0.8140382401i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2639307870 - 0.8140382401i\) |
\(L(1)\) |
\(\approx\) |
\(0.6419646282 - 0.4651811874i\) |
\(L(1)\) |
\(\approx\) |
\(0.6419646282 - 0.4651811874i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 67 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 - T \) |
| 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
−22.02022519168583213010024495044, −21.23991330855948460804163690634, −20.022529414486203418098995025272, −19.61810211839112373493346269350, −18.33610485128470500547476595495, −18.109302134808815231105407294066, −17.26644027095066766763170130485, −16.47191941325310329172507516142, −15.46170211207025807820858774160, −15.003396054732841569198983227160, −14.39949188324436025283068589735, −13.23471160964724766745870795938, −12.48637233101781009866461836922, −11.413553999726313290399887675165, −10.49151305736232160283025597680, −9.69095465024790703207159399535, −8.78065602989312987763541292802, −8.16269031082091989526776251363, −7.35970202848915083203144610824, −6.21794792973436975114753956324, −5.7055356861826991249998999349, −4.698316601293610804559560205424, −3.73798520358463958536137956400, −2.099256478720178743719186207925, −1.350426844776301745748997540,
0.23514288660044380612377333139, 1.102624330797520680842518000307, 2.14042345267422327810927215245, 3.22242327125988062301884995571, 4.21260704993458552770058250386, 4.76399657792741385861614364584, 6.389713779439913207107472785631, 7.14939273641520831560287150085, 8.25889839258445473253294013527, 8.83029909154230386801051276950, 9.71474978056121116861163560378, 10.72566422500767221209919906885, 11.25612016973508533943262358148, 11.8843680389405163291081490459, 13.02547594890409691510568632792, 13.85897142808846541563243755022, 14.17990950333597231262746751515, 15.690677151785407855360742381952, 16.59598556082062090866879744386, 17.07521877573717025457714068802, 18.02965813940786964661712586597, 18.62952876985764202590200517041, 19.59602303123059731898789486613, 20.0610562682498192807778414819, 20.98544910196766160351221012540