L(s) = 1 | + (−0.262 + 0.964i)2-s + (0.958 + 0.285i)3-s + (−0.861 − 0.506i)4-s + (−0.943 + 0.331i)5-s + (−0.527 + 0.849i)6-s + (0.926 + 0.377i)7-s + (0.715 − 0.698i)8-s + (0.836 + 0.548i)9-s + (−0.0724 − 0.997i)10-s + (−0.681 + 0.732i)11-s + (−0.681 − 0.732i)12-s + (−0.0724 + 0.997i)13-s + (−0.607 + 0.794i)14-s + (−0.998 + 0.0483i)15-s + (0.485 + 0.873i)16-s + (−0.906 − 0.421i)17-s + ⋯ |
L(s) = 1 | + (−0.262 + 0.964i)2-s + (0.958 + 0.285i)3-s + (−0.861 − 0.506i)4-s + (−0.943 + 0.331i)5-s + (−0.527 + 0.849i)6-s + (0.926 + 0.377i)7-s + (0.715 − 0.698i)8-s + (0.836 + 0.548i)9-s + (−0.0724 − 0.997i)10-s + (−0.681 + 0.732i)11-s + (−0.681 − 0.732i)12-s + (−0.0724 + 0.997i)13-s + (−0.607 + 0.794i)14-s + (−0.998 + 0.0483i)15-s + (0.485 + 0.873i)16-s + (−0.906 − 0.421i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4746884886 + 0.9273660449i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4746884886 + 0.9273660449i\) |
\(L(1)\) |
\(\approx\) |
\(0.7863425241 + 0.6765780780i\) |
\(L(1)\) |
\(\approx\) |
\(0.7863425241 + 0.6765780780i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 131 | \( 1 \) |
good | 2 | \( 1 + (-0.262 + 0.964i)T \) |
| 3 | \( 1 + (0.958 + 0.285i)T \) |
| 5 | \( 1 + (-0.943 + 0.331i)T \) |
| 7 | \( 1 + (0.926 + 0.377i)T \) |
| 11 | \( 1 + (-0.681 + 0.732i)T \) |
| 13 | \( 1 + (-0.0724 + 0.997i)T \) |
| 17 | \( 1 + (-0.906 - 0.421i)T \) |
| 19 | \( 1 + (0.120 + 0.992i)T \) |
| 23 | \( 1 + (-0.168 - 0.985i)T \) |
| 29 | \( 1 + (0.836 - 0.548i)T \) |
| 31 | \( 1 + (-0.989 + 0.144i)T \) |
| 37 | \( 1 + (0.995 - 0.0965i)T \) |
| 41 | \( 1 + (0.485 - 0.873i)T \) |
| 43 | \( 1 + (0.0241 + 0.999i)T \) |
| 47 | \( 1 + (-0.354 - 0.935i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (0.981 - 0.192i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
| 67 | \( 1 + (0.0241 - 0.999i)T \) |
| 71 | \( 1 + (0.885 + 0.464i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.568 - 0.822i)T \) |
| 83 | \( 1 + (0.215 + 0.976i)T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + (0.644 - 0.764i)T \) |
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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
−28.26890392795039194088579983591, −27.1142401711802252690233384124, −26.84004328291787406569875227996, −25.57355473834443508482988771600, −24.09178361764799395153996670854, −23.5964835524188471786457271875, −21.955478939670475310322977681615, −20.96725692595274750977620250075, −20.019148218422784120139900296557, −19.62792806381835707033110844264, −18.36537182705084733409369253551, −17.551209627213141655756603528288, −15.89317544222510192942551772829, −14.785790795644787712465421701726, −13.49728465457928390984615961640, −12.81347008124064462590936579509, −11.458047802067830721695260843370, −10.6190668716814164008258235721, −9.048025891914000675272215615817, −8.16656063145271757704130360848, −7.53111884156312444692361116903, −4.92414442582714049248640922684, −3.78523114260525264586734166549, −2.653439871046725931769358402208, −1.036303426644573767826912359568,
2.15849980317827096311636747476, 4.093334390547271788010381213294, 4.861730146508515533333800191571, 6.82626258451210226782205705649, 7.84065527028669614107425329238, 8.50847268828578647974138067041, 9.71565978127944685111439568475, 11.01014009558118450379505197303, 12.59269670719336032648834177526, 14.10500148951097149438709742874, 14.72234411712236910240211305387, 15.57967186983493135833826806737, 16.39332030289510231510437278064, 18.07075183453534676917264995654, 18.68971684525597881122709570366, 19.79961878013041993070037900460, 20.88603016109956909028210348426, 22.14216518597764979043872548738, 23.331052546156015822492998214378, 24.24734651686956595342747998882, 25.049471761608317601005336327444, 26.222310021697856182996491647444, 26.82948226557908365472393936193, 27.57347618090339794211429671280, 28.64669372498539591869002510654