L(s) = 1 | + (−0.247 + 0.968i)2-s + (0.203 − 0.979i)3-s + (−0.877 − 0.480i)4-s + (−0.998 + 0.0455i)5-s + (0.898 + 0.439i)6-s + (0.995 + 0.0909i)7-s + (0.682 − 0.730i)8-s + (−0.917 − 0.398i)9-s + (0.203 − 0.979i)10-s + (−0.917 − 0.398i)11-s + (−0.648 + 0.761i)12-s + (0.803 + 0.595i)13-s + (−0.334 + 0.942i)14-s + (−0.158 + 0.987i)15-s + (0.538 + 0.842i)16-s + (−0.775 + 0.631i)17-s + ⋯ |
L(s) = 1 | + (−0.247 + 0.968i)2-s + (0.203 − 0.979i)3-s + (−0.877 − 0.480i)4-s + (−0.998 + 0.0455i)5-s + (0.898 + 0.439i)6-s + (0.995 + 0.0909i)7-s + (0.682 − 0.730i)8-s + (−0.917 − 0.398i)9-s + (0.203 − 0.979i)10-s + (−0.917 − 0.398i)11-s + (−0.648 + 0.761i)12-s + (0.803 + 0.595i)13-s + (−0.334 + 0.942i)14-s + (−0.158 + 0.987i)15-s + (0.538 + 0.842i)16-s + (−0.775 + 0.631i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9911103305 + 0.2434583708i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9911103305 + 0.2434583708i\) |
\(L(1)\) |
\(\approx\) |
\(0.8268885408 + 0.1332929672i\) |
\(L(1)\) |
\(\approx\) |
\(0.8268885408 + 0.1332929672i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (-0.247 + 0.968i)T \) |
| 3 | \( 1 + (0.203 - 0.979i)T \) |
| 5 | \( 1 + (-0.998 + 0.0455i)T \) |
| 7 | \( 1 + (0.995 + 0.0909i)T \) |
| 11 | \( 1 + (-0.917 - 0.398i)T \) |
| 13 | \( 1 + (0.803 + 0.595i)T \) |
| 17 | \( 1 + (-0.775 + 0.631i)T \) |
| 19 | \( 1 + (0.746 + 0.665i)T \) |
| 23 | \( 1 + (0.854 - 0.519i)T \) |
| 29 | \( 1 + (-0.917 + 0.398i)T \) |
| 31 | \( 1 + (-0.974 - 0.225i)T \) |
| 37 | \( 1 + (0.803 + 0.595i)T \) |
| 41 | \( 1 + (-0.0682 - 0.997i)T \) |
| 43 | \( 1 + (0.934 - 0.356i)T \) |
| 47 | \( 1 + (-0.158 + 0.987i)T \) |
| 53 | \( 1 + (0.0227 + 0.999i)T \) |
| 59 | \( 1 + (0.0227 - 0.999i)T \) |
| 61 | \( 1 + (-0.829 + 0.557i)T \) |
| 67 | \( 1 + (0.682 + 0.730i)T \) |
| 71 | \( 1 + (0.962 + 0.269i)T \) |
| 73 | \( 1 + (0.0227 - 0.999i)T \) |
| 79 | \( 1 + (0.0227 - 0.999i)T \) |
| 83 | \( 1 + (0.898 + 0.439i)T \) |
| 89 | \( 1 + (0.291 - 0.956i)T \) |
| 97 | \( 1 + T \) |
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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
−21.343454523603346312986465504793, −20.87500253164473152326211622402, −20.0462130836822760206120748493, −19.86625312658430540827953167451, −18.43538045646857949466155088410, −18.04609666962924436981703855504, −17.03727192262343170083259820385, −16.06535390154674592832344736302, −15.36560066409341065871032999610, −14.660366062808910427157471345638, −13.570836137824943537018265649652, −12.88219228337559040154543570622, −11.58990969987037809833114312547, −11.13861160756776883667583369022, −10.7196801453425802951199428571, −9.52814418470795426940589161062, −8.83798844369484739806712791365, −7.98124385314837123565819569725, −7.43516549523569215640907379704, −5.30611554747758828227927019141, −4.85754072605022543222910803079, −3.95525654795243508827943061621, −3.171614140991197648143125398030, −2.23606827488619988525153062668, −0.708911079861569177187519279890,
0.82346082298586891235212298453, 1.89715225337818586228367288002, 3.38567367206054901109395906291, 4.39778275117949633023456146776, 5.434173211140261684829022908252, 6.26282326454587229665991778826, 7.36141829896579541279443773103, 7.7329727523852132949922778296, 8.56109404790368791569411307605, 9.03825920301805512563129321883, 10.8230610127384725213123801952, 11.19500659916678983558363114812, 12.408752932655431739883098866274, 13.16559207143891699119706047926, 14.014799381069995726881879299, 14.71701561045735189215731919561, 15.42275571859426981068988878575, 16.28727944770365814542795660384, 17.09092159023944841834588268422, 18.04359364458437949221005640460, 18.66935346675321304297281747668, 18.94932158345147915964342924373, 20.14892667033900580316924325178, 20.78916732659249326833691643165, 22.096341480842404426724039413647