L(s) = 1 | + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-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 + 35-s − 37-s + (−0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s + (0.5 + 0.866i)47-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-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 + 35-s − 37-s + (−0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s + (0.5 + 0.866i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.745677577 - 0.6353746767i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.745677577 - 0.6353746767i\) |
\(L(1)\) |
\(\approx\) |
\(1.263065016 - 0.2227124407i\) |
\(L(1)\) |
\(\approx\) |
\(1.263065016 - 0.2227124407i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 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
−31.19092655372331936658688498156, −30.42526831892212707758848705260, −29.436064620534546777552534944209, −28.318597120805601945142392832528, −26.9218358240940399710870186547, −26.138522546024668576217683024360, −25.11840314984425902353371873515, −23.57206135434059034606913410938, −22.89283692363934332560558352731, −21.43072641171321631759118402323, −20.61817670761550197646771554455, −19.14136138093604138617881530797, −18.02396237613425544392367353610, −17.12171379827086813108326308436, −15.621616933279643285793159534903, −14.28748392882596993623362620385, −13.578738511390696386748458350931, −11.830198563980962069457428171093, −10.59406520539960375815065004851, −9.641231071447980892461308889663, −7.741374952112187480015323770866, −6.746898685675789959251151415055, −5.08648366168555207668888162267, −3.42037687110345037306645455509, −1.63036367331278909056454026500,
1.101560726504607710923192621426, 2.96245566030996765290305857002, 5.06987474600708054945457132646, 5.87127898904817694011200059772, 8.019485911085724565308907214931, 8.89793845402069964832374016450, 10.35110236406855836961010352369, 11.838637757136705722088844444069, 12.93958269960061628301249253935, 14.1150714386743220817540831650, 15.5695588138305825180212761129, 16.59192368279084311297038268291, 17.89752281034223662205894727946, 18.82749525458060758769273809375, 20.47955372880888938086465315063, 21.1537796212054657106138292465, 22.30599009759026248389293879837, 23.81375358318902749690244378886, 24.72848694010798829289649574716, 25.547181956857525600145434504796, 27.07026158233903293823579314182, 28.10594312009296866917038589917, 28.92167657720444248972829447617, 30.12753641731710659504815192849, 31.40916438729701830231600969059