L(s) = 1 | + (−0.766 + 0.642i)5-s + (−0.766 − 0.642i)11-s + (−0.766 + 0.642i)13-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + (0.766 + 0.642i)29-s + (0.766 − 0.642i)31-s + 37-s + (−0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (0.766 + 0.642i)47-s + (−0.5 − 0.866i)53-s + 55-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)5-s + (−0.766 − 0.642i)11-s + (−0.766 + 0.642i)13-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + (0.766 + 0.642i)29-s + (0.766 − 0.642i)31-s + 37-s + (−0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (0.766 + 0.642i)47-s + (−0.5 − 0.866i)53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.050040736 - 0.08636968743i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.050040736 - 0.08636968743i\) |
\(L(1)\) |
\(\approx\) |
\(0.8949462433 + 0.02033281264i\) |
\(L(1)\) |
\(\approx\) |
\(0.8949462433 + 0.02033281264i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 11 | \( 1 + (-0.766 - 0.642i)T \) |
| 13 | \( 1 + (-0.766 + 0.642i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.939 + 0.342i)T \) |
| 29 | \( 1 + (0.766 + 0.642i)T \) |
| 31 | \( 1 + (0.766 - 0.642i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.766 + 0.642i)T \) |
| 43 | \( 1 + (0.939 - 0.342i)T \) |
| 47 | \( 1 + (0.766 + 0.642i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.173 + 0.984i)T \) |
| 61 | \( 1 + (-0.766 - 0.642i)T \) |
| 67 | \( 1 + (0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.939 - 0.342i)T \) |
| 83 | \( 1 + (0.766 + 0.642i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.939 - 0.342i)T \) |
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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
−22.69031764132102450585030674907, −21.4576424383489268591508244907, −20.81465640293848713207726850097, −20.037554262973374132544154272161, −19.279055313877676004296726832953, −18.565840636074190638472232857190, −17.35406053325366942708342554531, −16.913944969830252294927756170267, −15.80638309092145009238182358130, −15.22038167141519817877380060170, −14.45731501750874936522371932302, −13.1554900931028617641407370997, −12.49588939320423589606779501828, −11.99325168333595774847212388627, −10.67796690290755150784617797081, −10.10722788603953010898194393556, −8.94307005848624988088321833168, −7.99463228802874565887666962093, −7.55508642579575674164407834789, −6.2630830231297309527575720400, −5.139505908197421446174401205044, −4.48093371361693892407696186330, −3.38584392941519654647973628110, −2.26157150188645011251361432063, −0.87332723381316308490684913930,
0.69206042953961884899030378921, 2.525003344853591956818323243803, 3.0748503968885031031217366451, 4.35454964187152723403422974336, 5.14876644492015972734372629388, 6.43225439585168521004576363404, 7.23318361292040852166103969556, 7.96441903759052196412416030151, 8.98723777684048492384973832021, 9.97400354507083781251256905154, 10.96120351503651622421404523190, 11.51975996803195800581414130509, 12.4285366751230884957048158700, 13.459993505596638017441537640, 14.28171097241493945640727933777, 15.11205301617159300462619066740, 15.8447233166532382713920375511, 16.621185189813430252473320853897, 17.6007948223664638835813056802, 18.60136792389434805115011501568, 19.09015794656428557084275272302, 19.82864102566240426652984707451, 20.86811606848765526369185694549, 21.66807821883583138162046672913, 22.364891663827790373807932604079