L(s) = 1 | + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)11-s + 13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s − 41-s − 43-s + (−0.5 − 0.866i)47-s + (0.5 − 0.866i)53-s − 55-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)11-s + 13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s − 41-s − 43-s + (−0.5 − 0.866i)47-s + (0.5 − 0.866i)53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9967769186 + 0.3012402018i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9967769186 + 0.3012402018i\) |
\(L(1)\) |
\(\approx\) |
\(1.074967861 + 0.1837676773i\) |
\(L(1)\) |
\(\approx\) |
\(1.074967861 + 0.1837676773i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + T \) |
| 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 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 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 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + T \) |
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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
−30.70240701022664127605941315582, −29.57250083145304393826134011646, −28.54168642358002635848246799936, −27.793898664215444188694843991929, −26.33485601221573148492831030560, −25.44704753240674077354157787663, −24.217514882104440980528074946820, −23.51736342574846920645172794948, −21.88323939986568305355820530210, −21.09282213879159448654536614754, −20.03523669101452045643676811431, −18.74343109562507872501967674928, −17.56440889893560054951336434780, −16.47593214606831048913512484588, −15.50943266012893065845738897838, −13.799259918333861361662834776802, −13.119632906927244427614493881, −11.70602426091678641774294455588, −10.37608083676911194096067850761, −9.02124800570937866710137893216, −8.03934954937995374176232526471, −6.17895488106447636775583241984, −5.12199412191506097153837757958, −3.421243326092674281052487571783, −1.43379100056766816760335678131,
2.05185206543427806999833595911, 3.564680043885898841644550193032, 5.384314828234950571265997298464, 6.67307256621895801390059901298, 7.9111900103702069238136723198, 9.60860000931374621903166449591, 10.52936911897666048367824033756, 11.83691920232330516858510860407, 13.28592039162675572756403749578, 14.32265113883198674062414790567, 15.42726839865949797725185485791, 16.71169046993097783809968955934, 18.201901952651810769618017156568, 18.56681417231314503918701508207, 20.322027749755712132292159509471, 21.152274655726625266797544842406, 22.56515090861883152409931987495, 23.12427578416343378828026062659, 24.70474429028522404909683822056, 25.72911120558788813215907059910, 26.47489173174527488269798563393, 27.803961935896119077731209301355, 28.85471009034554244206355806159, 29.95026009123736014800726882183, 30.776101587770146938268129710393