L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.766 − 0.642i)3-s + (0.766 − 0.642i)4-s + (−0.642 + 0.766i)5-s + (−0.939 − 0.342i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s − 12-s + (−0.173 + 0.984i)13-s + (0.939 + 0.342i)14-s + (0.984 − 0.173i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + ⋯ |
L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.766 − 0.642i)3-s + (0.766 − 0.642i)4-s + (−0.642 + 0.766i)5-s + (−0.939 − 0.342i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s − 12-s + (−0.173 + 0.984i)13-s + (0.939 + 0.342i)14-s + (0.984 − 0.173i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.056721991 + 1.086983469i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.056721991 + 1.086983469i\) |
\(L(1)\) |
\(\approx\) |
\(1.477712692 + 0.001927544544i\) |
\(L(1)\) |
\(\approx\) |
\(1.477712692 + 0.001927544544i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 5 | \( 1 + (-0.642 + 0.766i)T \) |
| 7 | \( 1 + (0.766 + 0.642i)T \) |
| 11 | \( 1 + (0.342 + 0.939i)T \) |
| 13 | \( 1 + (-0.173 + 0.984i)T \) |
| 17 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (-0.766 + 0.642i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.984 + 0.173i)T \) |
| 31 | \( 1 + (0.984 + 0.173i)T \) |
| 41 | \( 1 + (0.342 - 0.939i)T \) |
| 43 | \( 1 + (-0.866 + 0.5i)T \) |
| 47 | \( 1 + (0.642 - 0.766i)T \) |
| 53 | \( 1 + (0.866 + 0.5i)T \) |
| 59 | \( 1 + (-0.173 - 0.984i)T \) |
| 61 | \( 1 + (-0.642 + 0.766i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (0.939 + 0.342i)T \) |
| 73 | \( 1 + (-0.766 - 0.642i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.642 + 0.766i)T \) |
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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
−18.106244509328474838722243311183, −17.14891312717301169278629353986, −16.919946983355014258710370417526, −16.40556334296979246824664726194, −15.51843551301354670060198925276, −15.11950045482736817811251222522, −14.44033827419450858378192667820, −13.46571809618130488071479953692, −12.970789738248525820871472511869, −12.08084313452346566870902378763, −11.577571413736126754987544084122, −11.0137080570084698945719598504, −10.44052299743011663129949415298, −9.27445785043180624850453258368, −8.45779995009358393321566863297, −7.7736605197403381334999184708, −7.08688908148911171796206765203, −6.1504601654571204565732135178, −5.381105738568451813006827771151, −4.90221444581911416810890597516, −4.29048649020386191935366001576, −3.57417728235055472951632662675, −2.87503798978155337548277831604, −1.30848150850309462779886191425, −0.55221863912994049660314687210,
1.2379482761751915868938439808, 1.940046842449400920928066027771, 2.51619108029002650730397561327, 3.693689117233398600018138968832, 4.39842134888569479900620326594, 5.04118996361618033901103025356, 5.89599037053085810081798066126, 6.52652184993851999814925067090, 7.20936299653499574440897342547, 7.72662589315365769041182562809, 8.76363885336201099383832392325, 9.97110849003178645112905467802, 10.63768107571501145694484054481, 11.23313186407627691634148874467, 11.97261550231737696221602871456, 12.13742227317788973856630939484, 12.88661471638009596239041915497, 13.84823348291501716134271744188, 14.56980154259689194958672836374, 14.95501109961072055691991065959, 15.58729790017638765173489836540, 16.64091375923517967516964781221, 17.09313832716605819962173261477, 18.08464451275088421491874734013, 18.827951893292287632202005717765