L(s) = 1 | + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + 17-s + 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s + 37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 − 0.866i)47-s − 53-s + 55-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + 17-s + 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s + 37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 − 0.866i)47-s − 53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.480086521 - 0.6901756796i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.480086521 - 0.6901756796i\) |
\(L(1)\) |
\(\approx\) |
\(1.033998219 - 0.1657419464i\) |
\(L(1)\) |
\(\approx\) |
\(1.033998219 - 0.1657419464i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 - T \) |
| 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 | \( 1 - T \) |
| 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
−25.89330310022770871160765057812, −25.16394566369824001285108219871, −23.8237661058980743546459672584, −23.26567781122369902396812222985, −22.2344186131216417912241151114, −21.45582948153370185147046351490, −20.28880127936821929385993009153, −19.38770169100170502320420558, −18.4210716456491971290320173698, −17.84328664592536056717421489023, −16.28840129758039107405566476667, −15.72751048235121027792125436882, −14.57060131154072672843021328106, −13.78405033624565219503436196758, −12.594909051472839162978750541667, −11.44631937894428746033878019950, −10.70591830838880919157371332679, −9.69173424107810420198163782963, −8.18232577702235877101476268223, −7.55794125200758817526452824367, −6.22187168447764165788327289192, −5.246196069779862567980022773531, −3.53413563213795278984493638362, −2.95130030518335197185953278806, −1.00902112903575942944947588110,
0.66913441859890203678422779921, 2.08665297668443450188056742711, 3.748898509142854817555213786473, 4.70428668362548930345766017071, 5.80273663414171062844313565361, 7.26770196451346934590789314351, 8.12327000235470804977367086064, 9.234621339431881432943121456168, 10.16690633403022728487410864594, 11.58829145254868960072789261092, 12.25955061538853320094514797315, 13.2671712421355312487799375958, 14.33211723927893636889228505908, 15.518028911273590787519264457882, 16.27211577073849139361872933207, 17.12119459423698867921076095632, 18.331544639534929639123967976183, 19.1396855003431497710946186885, 20.451363020596059903763283516890, 20.68005103204805613626227853612, 21.97958697949751069572453640561, 23.14473161581770401837288826632, 23.736849494462215422005379810634, 24.67854870923453552436600875957, 25.63773372814071554343610270992