L(s) = 1 | + (−0.142 − 0.989i)3-s + (−0.841 + 0.540i)5-s + (−0.841 + 0.540i)7-s + (−0.959 + 0.281i)9-s + (0.841 + 0.540i)11-s + (0.142 + 0.989i)13-s + (0.654 + 0.755i)15-s + (−0.654 + 0.755i)17-s + (−0.959 + 0.281i)19-s + (0.654 + 0.755i)21-s + (0.959 − 0.281i)23-s + (0.415 − 0.909i)25-s + (0.415 + 0.909i)27-s + (−0.841 + 0.540i)29-s + (0.959 + 0.281i)31-s + ⋯ |
L(s) = 1 | + (−0.142 − 0.989i)3-s + (−0.841 + 0.540i)5-s + (−0.841 + 0.540i)7-s + (−0.959 + 0.281i)9-s + (0.841 + 0.540i)11-s + (0.142 + 0.989i)13-s + (0.654 + 0.755i)15-s + (−0.654 + 0.755i)17-s + (−0.959 + 0.281i)19-s + (0.654 + 0.755i)21-s + (0.959 − 0.281i)23-s + (0.415 − 0.909i)25-s + (0.415 + 0.909i)27-s + (−0.841 + 0.540i)29-s + (0.959 + 0.281i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02743137032 + 0.1173592462i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02743137032 + 0.1173592462i\) |
\(L(1)\) |
\(\approx\) |
\(0.6749661780 + 0.04001585562i\) |
\(L(1)\) |
\(\approx\) |
\(0.6749661780 + 0.04001585562i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 7 | \( 1 + (-0.841 + 0.540i)T \) |
| 11 | \( 1 + (0.841 + 0.540i)T \) |
| 13 | \( 1 + (0.142 + 0.989i)T \) |
| 17 | \( 1 + (-0.654 + 0.755i)T \) |
| 19 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (0.959 - 0.281i)T \) |
| 29 | \( 1 + (-0.841 + 0.540i)T \) |
| 31 | \( 1 + (0.959 + 0.281i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.142 + 0.989i)T \) |
| 43 | \( 1 + (0.841 + 0.540i)T \) |
| 47 | \( 1 + (0.142 - 0.989i)T \) |
| 53 | \( 1 + (0.142 + 0.989i)T \) |
| 59 | \( 1 + (-0.142 + 0.989i)T \) |
| 61 | \( 1 + (-0.415 - 0.909i)T \) |
| 67 | \( 1 + (-0.142 - 0.989i)T \) |
| 71 | \( 1 + (-0.841 - 0.540i)T \) |
| 73 | \( 1 + (-0.959 - 0.281i)T \) |
| 79 | \( 1 + (0.959 + 0.281i)T \) |
| 83 | \( 1 + (-0.654 + 0.755i)T \) |
| 97 | \( 1 + (0.841 - 0.540i)T \) |
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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
−22.219853199330818337804737461450, −20.84512006157285054244168761105, −20.49286787966130163005010203470, −19.47133198915724520329098340698, −19.12631798368697734586250292031, −17.339778981227635228734980995660, −17.07278548197813104292786458640, −15.98264646588078647068643369917, −15.640935556730022575853599872302, −14.75357893965709582812831388434, −13.60785523293029569212221576746, −12.81203127279088326221040154728, −11.74733039417878286285680833730, −11.03965526736430258718852346560, −10.23736146116251869256336570096, −9.131777954192478197923342637978, −8.70265536706206846207534628108, −7.460816531874815718205081947476, −6.42518206026278702712170971980, −5.38966551032523732068312976629, −4.35089305203985989326382824588, −3.711683142302137148299574589876, −2.85450542793223249116174384092, −0.77520284760991437500208923120, −0.04002402883633571086289489668,
1.50303526686189401945398344889, 2.51560756326404431458812124913, 3.574493393871361374923242584785, 4.58785746114542521882205750832, 6.18389651918930347328308795099, 6.6138165350171517225196032084, 7.326691296207517214665048922429, 8.54455819680074503723245907861, 9.10832148324656874106220746874, 10.52818844187707111724615240308, 11.37731885881101566663983931117, 12.1823614726869578533566466024, 12.69447967909889692381530573616, 13.71707903893377874922565704876, 14.739064998049416547101385417877, 15.27943120436841870575686154762, 16.490714771890801136337666960494, 17.11822992333003007076760779045, 18.20151588361718526540948313170, 18.99380306570253641699171489593, 19.36083297038542381700460543605, 20.029395604243943235792470881418, 21.38408212499191991958756135462, 22.350554486702785205189543187525, 22.89512903497118842667622776949