L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.866 − 0.5i)3-s + (−0.978 − 0.207i)4-s + (−0.669 + 0.743i)5-s + (−0.587 + 0.809i)6-s + (−0.309 + 0.951i)8-s + (0.5 + 0.866i)9-s + (0.669 + 0.743i)10-s + (−0.743 + 0.669i)11-s + (0.743 + 0.669i)12-s + (0.587 − 0.809i)13-s + (0.951 − 0.309i)15-s + (0.913 + 0.406i)16-s + (0.743 − 0.669i)17-s + (0.913 − 0.406i)18-s + (0.406 − 0.913i)19-s + ⋯ |
L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.866 − 0.5i)3-s + (−0.978 − 0.207i)4-s + (−0.669 + 0.743i)5-s + (−0.587 + 0.809i)6-s + (−0.309 + 0.951i)8-s + (0.5 + 0.866i)9-s + (0.669 + 0.743i)10-s + (−0.743 + 0.669i)11-s + (0.743 + 0.669i)12-s + (0.587 − 0.809i)13-s + (0.951 − 0.309i)15-s + (0.913 + 0.406i)16-s + (0.743 − 0.669i)17-s + (0.913 − 0.406i)18-s + (0.406 − 0.913i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5449649596 - 0.4546446294i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5449649596 - 0.4546446294i\) |
\(L(1)\) |
\(\approx\) |
\(0.6151726574 - 0.3480190315i\) |
\(L(1)\) |
\(\approx\) |
\(0.6151726574 - 0.3480190315i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-0.669 + 0.743i)T \) |
| 11 | \( 1 + (-0.743 + 0.669i)T \) |
| 13 | \( 1 + (0.587 - 0.809i)T \) |
| 17 | \( 1 + (0.743 - 0.669i)T \) |
| 19 | \( 1 + (0.406 - 0.913i)T \) |
| 23 | \( 1 + (-0.104 + 0.994i)T \) |
| 29 | \( 1 + (-0.951 + 0.309i)T \) |
| 31 | \( 1 + (0.669 + 0.743i)T \) |
| 37 | \( 1 + (0.669 - 0.743i)T \) |
| 43 | \( 1 + (0.809 + 0.587i)T \) |
| 47 | \( 1 + (0.994 + 0.104i)T \) |
| 53 | \( 1 + (0.207 - 0.978i)T \) |
| 59 | \( 1 + (0.913 - 0.406i)T \) |
| 61 | \( 1 + (-0.913 - 0.406i)T \) |
| 67 | \( 1 + (-0.207 + 0.978i)T \) |
| 71 | \( 1 + (0.951 + 0.309i)T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.866 - 0.5i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.406 + 0.913i)T \) |
| 97 | \( 1 + (0.951 - 0.309i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.84850338620479433457458351624, −24.447208876761775920321217510237, −23.91848799848116487328039185436, −23.22850770584845639766558847448, −22.42629706654194777464148637857, −21.29018373873407787135446988495, −20.6843483257411938399568454648, −18.89349332474547077108528618320, −18.424101061520601528220489380294, −16.86303186824986084322804470253, −16.67637188886548336444484385135, −15.8010749232092648072315140072, −15.02878384056625535441935077755, −13.75832910833102392739658613434, −12.66896496953795132949024061421, −11.877521930995236614975978276031, −10.66168960400515102867538313741, −9.5018119141284181814189893124, −8.46239043556711076950183408547, −7.5948527687931061956907872368, −6.17950087796733698580158121548, −5.5217436530651229571288160736, −4.38784937218387349783127465834, −3.67234015863215428730378235088, −0.86738957173492508055880768808,
0.80760278721895773138599432335, 2.38246645243484250989617173607, 3.47369038176406752163319478234, 4.83735288576650678576389260974, 5.72945274236388978256710426664, 7.23365086380231315143776547375, 7.974131056528900138194367130913, 9.63287314256454813476164605193, 10.64239354405785143034142075325, 11.243231801715715023942108673228, 12.12608362544357182589555873245, 12.97124224503985499759557810080, 13.88369825684378236549852813487, 15.18096667271639842288579645729, 16.1112514993321910847780390831, 17.70835556616814228454210575008, 18.046050985169346027195986963005, 18.93626178054851069250513976682, 19.77548956296788846558402666182, 20.7911818261659191611430926145, 21.89241629999442885974274867584, 22.73044144320214507968170636596, 23.227675227032458580252180170190, 23.92463116190039369454805728905, 25.402649619903536290696434738788