L(s) = 1 | + (−0.644 − 0.764i)2-s + (−0.584 − 0.811i)3-s + (−0.169 + 0.985i)4-s + (0.0567 + 0.998i)5-s + (−0.243 + 0.969i)6-s + (−0.914 + 0.404i)7-s + (0.862 − 0.505i)8-s + (−0.316 + 0.948i)9-s + (0.726 − 0.686i)10-s + (−0.0189 − 0.999i)11-s + (0.898 − 0.438i)12-s + (0.421 − 0.906i)13-s + (0.898 + 0.438i)14-s + (0.776 − 0.629i)15-s + (−0.942 − 0.334i)16-s + (0.132 − 0.991i)17-s + ⋯ |
L(s) = 1 | + (−0.644 − 0.764i)2-s + (−0.584 − 0.811i)3-s + (−0.169 + 0.985i)4-s + (0.0567 + 0.998i)5-s + (−0.243 + 0.969i)6-s + (−0.914 + 0.404i)7-s + (0.862 − 0.505i)8-s + (−0.316 + 0.948i)9-s + (0.726 − 0.686i)10-s + (−0.0189 − 0.999i)11-s + (0.898 − 0.438i)12-s + (0.421 − 0.906i)13-s + (0.898 + 0.438i)14-s + (0.776 − 0.629i)15-s + (−0.942 − 0.334i)16-s + (0.132 − 0.991i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3188120549 - 0.4382452862i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3188120549 - 0.4382452862i\) |
\(L(1)\) |
\(\approx\) |
\(0.5165292633 - 0.2954482504i\) |
\(L(1)\) |
\(\approx\) |
\(0.5165292633 - 0.2954482504i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 167 | \( 1 \) |
good | 2 | \( 1 + (-0.644 - 0.764i)T \) |
| 3 | \( 1 + (-0.584 - 0.811i)T \) |
| 5 | \( 1 + (0.0567 + 0.998i)T \) |
| 7 | \( 1 + (-0.914 + 0.404i)T \) |
| 11 | \( 1 + (-0.0189 - 0.999i)T \) |
| 13 | \( 1 + (0.421 - 0.906i)T \) |
| 17 | \( 1 + (0.132 - 0.991i)T \) |
| 19 | \( 1 + (0.988 - 0.150i)T \) |
| 23 | \( 1 + (0.614 - 0.788i)T \) |
| 29 | \( 1 + (0.614 + 0.788i)T \) |
| 31 | \( 1 + (-0.387 - 0.922i)T \) |
| 37 | \( 1 + (-0.316 - 0.948i)T \) |
| 41 | \( 1 + (-0.700 + 0.713i)T \) |
| 43 | \( 1 + (0.974 - 0.225i)T \) |
| 47 | \( 1 + (0.351 - 0.936i)T \) |
| 53 | \( 1 + (-0.0944 + 0.995i)T \) |
| 59 | \( 1 + (0.132 + 0.991i)T \) |
| 61 | \( 1 + (0.822 - 0.569i)T \) |
| 67 | \( 1 + (0.0567 - 0.998i)T \) |
| 71 | \( 1 + (0.553 - 0.832i)T \) |
| 73 | \( 1 + (-0.942 + 0.334i)T \) |
| 79 | \( 1 + (-0.521 - 0.853i)T \) |
| 83 | \( 1 + (-0.644 + 0.764i)T \) |
| 89 | \( 1 + (0.776 + 0.629i)T \) |
| 97 | \( 1 + (-0.387 + 0.922i)T \) |
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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
−27.89114162937091002126425529890, −26.980019812250952101976757163488, −25.98790729307011208456784421017, −25.332065123510517804846595976796, −23.89488417502656464787143675859, −23.34030623220327091513601737063, −22.35216454531452157286923290186, −20.99158489564868658588545872615, −20.08707230334427688630887923441, −19.09184085374981046009312472362, −17.554741124193379781485666222344, −17.064401620563538244132147872210, −16.04529698388146674854402071773, −15.61416348272098327354982099277, −14.20318096471615037676742921730, −12.9104275605649211095181131597, −11.64514633808027338611739954918, −10.2016891497880403680290931175, −9.58580251020324617588886347227, −8.685664212426673230785922506398, −7.13779340442374099664883253862, −6.04896318002091832546881797298, −4.97480080001324987058730203799, −3.92599046705267753893908962066, −1.28291235605041336456920756475,
0.688825372133209256239735653952, 2.58807379571740181615895996314, 3.28608835832414354069802531167, 5.56320115986948741987038702874, 6.73369217052010413096889586116, 7.67689714756059528809192956801, 8.98868044430001536915919413128, 10.32040443405823589603775875346, 11.11158157587789684248755916611, 12.0334354348827240015747088518, 13.10669463585770096633492698239, 13.92714824809725353157246343861, 15.807047612507102542968899825, 16.67653138427286689959310750311, 18.02202516640070781301856819215, 18.488054250578851308524303112425, 19.20066270354592920753458633383, 20.22840149980238223852350600904, 21.74185609073088389680509796705, 22.45666948430605047855331523557, 23.05747233581535018449566502969, 24.746850138810244238651699132635, 25.48127952482998046486567003000, 26.52385496210557511964487320855, 27.42906320136291868386937001554