L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 13-s + 15-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 − 27-s − 29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 13-s + 15-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 − 27-s − 29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.538079717 + 0.7624680365i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.538079717 + 0.7624680365i\) |
\(L(1)\) |
\(\approx\) |
\(1.272844167 + 0.08827015491i\) |
\(L(1)\) |
\(\approx\) |
\(1.272844167 + 0.08827015491i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 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 | \( 1 + (-0.5 - 0.866i)T \) |
| 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 | \( 1 + T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.08944930441852046345594614970, −20.40079230282130746342757296711, −20.09645384162736883692991797461, −18.80294815555291664736462461723, −18.231631244306730471197719359733, −16.981609229234316878182828086216, −16.54661191214029992130639383908, −15.746044882935623892616108737783, −15.20412125897820017071595346902, −13.872160671387444000364022104928, −13.60984948038648243688900642716, −12.83256329636133300837997192747, −11.45897555408821010499808532219, −10.94792718500418358928129812046, −9.94606313266839916028844639346, −9.10986851914178723671097920510, −8.68085259380252604327888862618, −7.84685903068439254395477473045, −6.5317181539228640477926404638, −5.43786650200604149252701489624, −4.96179966861733440862807213132, −3.92621765341618320939427380776, −3.003764913561583665675137761292, −2.0535818975590507988566872607, −0.63262202838960102264971423455,
1.505690533216996031706094673547, 2.023658838595023675338400886611, 3.17562464748980552560272518699, 3.83557843285783484441750594970, 5.45054308753465703670618833057, 6.128586641249302114443092405048, 7.06482246008139492243736290549, 7.58274848751537989027653170989, 8.60491032052114770612973781915, 9.429096064524144461759739917921, 10.37628026319877772391044285165, 11.12558237936639837787670629195, 12.10277605868317863230212834225, 13.05496983957448226906385391163, 13.46693893128170748569782556283, 14.4277882933334702357303429835, 14.97600100841294603803745976504, 15.79057137088339903602597241472, 17.037860204492732089658757064975, 17.934942022736359130538292986873, 18.18098803455023703922486127643, 19.03971003666964556144325451483, 19.74283090711064134537908225341, 20.69338930339328367067297933476, 21.20452182463848047803919823093