L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.939 + 0.342i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.766 − 0.642i)6-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (0.939 − 0.342i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.766 − 0.642i)13-s + (−0.939 + 0.342i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + (−0.5 − 0.866i)18-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.939 + 0.342i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.766 − 0.642i)6-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (0.939 − 0.342i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.766 − 0.642i)13-s + (−0.939 + 0.342i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + (−0.5 − 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04382980453 + 0.4216297389i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04382980453 + 0.4216297389i\) |
\(L(1)\) |
\(\approx\) |
\(0.6411558106 + 0.1569700196i\) |
\(L(1)\) |
\(\approx\) |
\(0.6411558106 + 0.1569700196i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (0.939 + 0.342i)T \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.766 - 0.642i)T \) |
| 17 | \( 1 + (-0.766 + 0.642i)T \) |
| 23 | \( 1 + (0.173 - 0.984i)T \) |
| 29 | \( 1 + (0.173 - 0.984i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.766 + 0.642i)T \) |
| 43 | \( 1 + (-0.939 - 0.342i)T \) |
| 47 | \( 1 + (-0.766 - 0.642i)T \) |
| 53 | \( 1 + (0.766 + 0.642i)T \) |
| 59 | \( 1 + (-0.766 + 0.642i)T \) |
| 61 | \( 1 + (-0.173 + 0.984i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (-0.939 - 0.342i)T \) |
| 73 | \( 1 + (0.939 + 0.342i)T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.939 - 0.342i)T \) |
| 97 | \( 1 + (-0.173 - 0.984i)T \) |
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
−27.52911433201416360322306823889, −26.890142332549039367771339802, −26.06973815142951163177731022324, −24.904510581306276771380619530200, −24.1897488012504752416953582893, −23.54598351835968101853543861885, −21.51763060947553110552108239013, −20.41035226611596374588995403106, −19.64163562647820293692290358889, −18.957769135852510454946137401891, −17.899712209653584043550032992917, −16.50385525458439057509788458350, −15.75220370224763215632199388650, −14.72918148741363596930833519135, −13.527216710445379455129030347986, −12.16746492612627686029925726776, −11.028078691793739380986234339391, −9.436890076984964346320125379033, −8.74243736104965033702059752134, −7.73890079887266694013758252844, −6.89061136657828010997201712243, −5.075048021881858321115199862588, −3.29531355013923491075424913261, −1.77114226094641901412150743744, −0.18979522857209789804288369775,
2.13205161672140158883528738933, 3.11721472130649358135632598759, 4.42836531379083298747626725714, 6.86964939675613359910902243267, 7.760979583960537163782469152760, 8.62410347826466828358851709312, 10.018349433118266910297089984661, 10.60331344267383058614657754348, 12.03347239195690623952098451261, 13.155453328482348816068517715234, 15.04662808717695325466015733525, 15.22795413214450813067324548766, 16.58948436077633288091213273881, 17.95510079698802317215624932906, 18.84812385066013514485781305123, 19.84974709502560568847360473939, 20.3197723632573806773129625475, 21.588284354561589135293890150935, 22.59355581091162557038017500584, 24.17835137564604101899255688415, 25.22813652006436110637912826406, 26.1753492558387388637611048782, 26.78169094301070052077879442749, 27.59267118589589598681248895599, 28.61170166260997103494014257375