L(s) = 1 | + (−0.994 − 0.104i)2-s + (−0.978 + 0.207i)3-s + (0.978 + 0.207i)4-s + (−0.587 − 0.809i)5-s + (0.994 − 0.104i)6-s + (−0.207 + 0.978i)7-s + (−0.951 − 0.309i)8-s + (0.913 − 0.406i)9-s + (0.5 + 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (0.743 + 0.669i)15-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (−0.951 + 0.309i)18-s + (−0.743 + 0.669i)19-s + ⋯ |
L(s) = 1 | + (−0.994 − 0.104i)2-s + (−0.978 + 0.207i)3-s + (0.978 + 0.207i)4-s + (−0.587 − 0.809i)5-s + (0.994 − 0.104i)6-s + (−0.207 + 0.978i)7-s + (−0.951 − 0.309i)8-s + (0.913 − 0.406i)9-s + (0.5 + 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (0.743 + 0.669i)15-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (−0.951 + 0.309i)18-s + (−0.743 + 0.669i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1847725716 - 0.2360753186i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1847725716 - 0.2360753186i\) |
\(L(1)\) |
\(\approx\) |
\(0.4196514215 + 0.02149661258i\) |
\(L(1)\) |
\(\approx\) |
\(0.4196514215 + 0.02149661258i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.994 - 0.104i)T \) |
| 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 5 | \( 1 + (-0.587 - 0.809i)T \) |
| 7 | \( 1 + (-0.207 + 0.978i)T \) |
| 17 | \( 1 + (0.104 + 0.994i)T \) |
| 19 | \( 1 + (-0.743 + 0.669i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.669 - 0.743i)T \) |
| 31 | \( 1 + (0.587 - 0.809i)T \) |
| 37 | \( 1 + (-0.743 - 0.669i)T \) |
| 41 | \( 1 + (-0.207 - 0.978i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.951 - 0.309i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.207 + 0.978i)T \) |
| 61 | \( 1 + (-0.104 - 0.994i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.994 - 0.104i)T \) |
| 73 | \( 1 + (0.951 - 0.309i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.587 - 0.809i)T \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.406 + 0.913i)T \) |
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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
−28.22305105727009511348890188159, −27.21987771995118626498010596687, −26.71482394893286099679183746201, −25.6010176305459499262429413876, −24.33092940029342645187206326862, −23.39551765047414673664337569340, −22.707549490680497236846803227564, −21.340923330507721721607508936388, −20.00382070748171461961677909410, −19.11593938575850369170605356011, −18.225687680254267695881987135347, −17.350112061936121052840688312173, −16.3833257974911225130022611135, −15.58529810271846663638741670115, −14.21622082137718129236398713235, −12.59384142738515186635599182390, −11.37743052446406896019474631100, −10.754729273592940003757513266481, −9.86348317331947745276045015915, −8.13984616085317312629101945697, −6.92607547297033622198549855803, −6.60452300158633920342847347160, −4.70767284129839924668492494576, −2.92487879809748750559088993298, −0.98408047879850427757185844806,
0.23319809803882715851285150311, 1.762154362237056654708304394157, 3.78660531816340635894292166214, 5.40942231877068626283383736226, 6.419457563070023542568335917422, 7.90614899534129844425667303014, 8.92215895522804817549823063517, 10.00140634482994180236867599977, 11.19699902168793107218870812697, 12.11604914275673433151337504312, 12.73522413788353408955118933840, 15.248614884290447352031963400334, 15.75131682190366513850712482823, 16.86081106634031403916747466912, 17.4855477517136894017691393035, 18.81517243540489106849014084149, 19.41149319539505195145005662057, 20.921534552367790985767235435981, 21.500579487684371694582294135872, 22.90730268095818376946770419445, 24.0057809350817753918852423798, 24.79293399741106362490903538542, 25.91604504407481648397346573474, 27.26202547087101972009986448371, 27.79209344755342839384547502970