L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + 11-s + 12-s + 14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + 18-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + 11-s + 12-s + 14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + 18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1224247765 - 0.3460310091i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1224247765 - 0.3460310091i\) |
\(L(1)\) |
\(\approx\) |
\(0.3982294028 - 0.3611193758i\) |
\(L(1)\) |
\(\approx\) |
\(0.3982294028 - 0.3611193758i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 79 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 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 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 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
−22.336448599759245745359691794802, −21.589079768157577489988893780345, −20.25630210916230104577083809406, −19.505507240437085350413208833251, −19.07700178115768280729533055330, −17.688660067662221087905626222963, −17.37710686105963878056216013801, −16.59338648833427279704013220645, −15.77506117416946489484004691215, −15.19547027821940489243888871546, −14.47861590129481712700395940914, −13.75450537385609588510651569010, −12.51094951955189075272164502401, −11.24403317503259020666291056965, −10.7233595161925308264772903898, −10.10273856265062282940800979002, −9.14615912158574534400760350908, −8.4386417387556755338742253373, −7.01223653984193377795935117371, −6.77106244031290235836208282510, −5.9193836725232419451495889282, −4.59247688592934330605088643961, −4.06886708323597637974920468449, −3.04703352629600317836274934051, −1.13759011416925982918846057315,
0.24198689204063049279303105195, 1.3386638270328411564498311043, 2.221534293633340079620881718, 3.323198225136784615772899793134, 4.442946947709626187213821713492, 5.348565203547375697869526244328, 6.473316417455602677998072071679, 7.38127089989200515320563240530, 8.39104935237622433738360353821, 8.92779105838621728442400774073, 9.72777445086274085187861344986, 11.01773426527712078255759728512, 11.72329505737781948211751911127, 12.2093401680351459167572737695, 12.85859635320968611729222498690, 13.52965798615151260665866572504, 14.708597058758830762014455544196, 16.06819244425477443746668199337, 16.56358827581496083843388477051, 17.36545637721611073958088902861, 18.093666655487452808566110385692, 19.05310237445316192109143446832, 19.32045442653645406893886630896, 20.12813780712958822518182830890, 20.97921289327010677381371975549