L(s) = 1 | + (0.173 − 0.984i)3-s + (−0.173 + 0.984i)5-s + (−0.939 − 0.342i)9-s − 11-s + (−0.766 + 0.642i)13-s + (0.939 + 0.342i)15-s + (0.939 − 0.342i)17-s + (−0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (0.173 + 0.984i)29-s + (−0.5 + 0.866i)31-s + (−0.173 + 0.984i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)3-s + (−0.173 + 0.984i)5-s + (−0.939 − 0.342i)9-s − 11-s + (−0.766 + 0.642i)13-s + (0.939 + 0.342i)15-s + (0.939 − 0.342i)17-s + (−0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (0.173 + 0.984i)29-s + (−0.5 + 0.866i)31-s + (−0.173 + 0.984i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3341112515 + 0.4465675039i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3341112515 + 0.4465675039i\) |
\(L(1)\) |
\(\approx\) |
\(0.7947170086 + 0.01511877040i\) |
\(L(1)\) |
\(\approx\) |
\(0.7947170086 + 0.01511877040i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.173 - 0.984i)T \) |
| 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.766 + 0.642i)T \) |
| 17 | \( 1 + (0.939 - 0.342i)T \) |
| 23 | \( 1 + (-0.766 + 0.642i)T \) |
| 29 | \( 1 + (0.173 + 0.984i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.766 - 0.642i)T \) |
| 43 | \( 1 + (0.939 - 0.342i)T \) |
| 47 | \( 1 + (-0.939 - 0.342i)T \) |
| 53 | \( 1 + (0.173 + 0.984i)T \) |
| 59 | \( 1 + (-0.939 + 0.342i)T \) |
| 61 | \( 1 + (-0.766 + 0.642i)T \) |
| 67 | \( 1 + (-0.766 + 0.642i)T \) |
| 71 | \( 1 + (0.939 - 0.342i)T \) |
| 73 | \( 1 + (-0.173 + 0.984i)T \) |
| 79 | \( 1 + (0.939 - 0.342i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.173 - 0.984i)T \) |
| 97 | \( 1 + (-0.173 + 0.984i)T \) |
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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
−23.13635615548265790460149767337, −22.35276593815778812489640632557, −21.26433358826759352237996135421, −20.856836724049290284156455321481, −20.02260767485382428438846143147, −19.29702355374600577834225105617, −18.02427277730117673096652591027, −17.02735371258203868832154833824, −16.417808251596566939449164958668, −15.6090033051494643969865758790, −14.89858138247790372127319521088, −13.87258589561899328547795611940, −12.82335530237517467392223217027, −12.092576635542165144164992925044, −10.93642388149731859799210477501, −10.015614174025489264671002634819, −9.425317978844418978524365930742, −8.13892493445321917077485352527, −7.84896247089335969117175957207, −5.92124629582151855726424451455, −5.17570466450450859736605556296, −4.392282028716073497450170566885, −3.31741021882467739505425727710, −2.15638278361179038067459316672, −0.26940712178690825956965508514,
1.64060623077694257378902548756, 2.6809393632264763830460624496, 3.45427027636875505786533133205, 5.09010675982363277446748597387, 6.08947910860738039541827854366, 7.21181226071302751814837430543, 7.52471060468110072647389515946, 8.65760574311611899703458891196, 9.89445648707640939501355630105, 10.762945326352292959480078353777, 11.84758575599282050036722381223, 12.39540292695141024757704047572, 13.635221072605690932997150463340, 14.18304556877127292927375595191, 15.00985469065378805243942605659, 16.04863394843515792181003128531, 17.14978104761329480687511460571, 18.11677724001504562513346123291, 18.59501141006419770091123458710, 19.35550268084467620033566887558, 20.13905680881977224174807036056, 21.269198356073669358969126850175, 22.11612288798838908642390583846, 23.102624608861365442168850401425, 23.67445059497828356596994465622