L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + 6-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s − 13-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + 6-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s − 13-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7525569845 - 0.1762949633i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7525569845 - 0.1762949633i\) |
\(L(1)\) |
\(\approx\) |
\(0.6353170310 + 0.04031733613i\) |
\(L(1)\) |
\(\approx\) |
\(0.6353170310 + 0.04031733613i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 167 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + T \) |
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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
−21.21027325478898817585071613366, −20.04154556844217069473170702402, −19.31147289052059806703840099175, −18.66707592621895082375215709800, −17.950834124901479558404489235998, −16.98975706509279423933835091763, −16.81216710281008047443829949520, −16.09372595431462333389972257873, −14.83118045861352297120888022356, −14.17818776505995314338216744067, −13.23702267615201062098485928978, −12.75017251756102786123037603842, −11.829274641156256908638063774152, −10.59255476015966047971502459094, −10.073298954039653993296192612067, −8.85707038527822360027911061181, −8.24782167401486953809816891477, −7.60248914858776361207149567158, −6.51416496910188078736784137887, −5.836151628299670730442377689529, −5.26086796265586070900396873857, −4.31776602724321882658356404287, −2.49716198626592566371149323734, −1.47447707606016446843359042096, −0.618654355908734090544029348259,
0.31987755096635941552143140526, 1.8331206662793982464701083121, 2.83341351621336777931202739711, 3.41325514883346243626261207240, 4.84621364645989945999755021822, 5.084692467830468831392455189725, 6.69732890227240935187516512369, 7.28786891201200649636947299481, 8.51974383746314033759291905636, 9.708154218741172964909211932021, 9.81380145208510466650081886251, 10.62453147298262774304279111815, 11.43415464604829629645128529183, 12.05453446418173927248973777016, 13.03374575562795405877819402808, 13.96940357517773301253772846260, 14.88770811037707305380534778946, 15.55767867712869177342781809722, 16.67013187451852973856399565719, 17.33563305907067500830202081560, 17.91764846753657115265661830125, 18.53948562711846671139560422572, 19.632110108794448629753705895216, 20.212383861684808191520837229021, 21.25359816003052937360882327961