L(s) = 1 | + (−1.87 + 0.682i)2-s + (−0.0840 + 0.476i)3-s + (2.28 − 1.91i)4-s + (−0.167 − 0.950i)6-s + (0.848 + 1.46i)7-s + (−1.97 + 3.42i)8-s + (0.719 + 0.261i)9-s + (0.5 − 0.866i)11-s + (0.721 + 1.24i)12-s + (−0.173 − 0.984i)13-s + (−2.59 − 2.17i)14-s + (0.851 − 4.82i)16-s − 1.52·18-s + (0.719 + 0.694i)19-s + (−0.771 + 0.280i)21-s + (−0.346 + 1.96i)22-s + ⋯ |
L(s) = 1 | + (−1.87 + 0.682i)2-s + (−0.0840 + 0.476i)3-s + (2.28 − 1.91i)4-s + (−0.167 − 0.950i)6-s + (0.848 + 1.46i)7-s + (−1.97 + 3.42i)8-s + (0.719 + 0.261i)9-s + (0.5 − 0.866i)11-s + (0.721 + 1.24i)12-s + (−0.173 − 0.984i)13-s + (−2.59 − 2.17i)14-s + (0.851 − 4.82i)16-s − 1.52·18-s + (0.719 + 0.694i)19-s + (−0.771 + 0.280i)21-s + (−0.346 + 1.96i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6173370983\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6173370983\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (-0.719 - 0.694i)T \) |
good | 2 | \( 1 + (1.87 - 0.682i)T + (0.766 - 0.642i)T^{2} \) |
| 3 | \( 1 + (0.0840 - 0.476i)T + (-0.939 - 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.848 - 1.46i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (0.943 - 0.791i)T + (0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.0121 - 0.0687i)T + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (-1.47 + 1.23i)T + (0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.280 + 1.59i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (0.374 + 0.648i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.158535649190805633750899850842, −8.516483066296914308343205323684, −7.901179316542296920787914519406, −7.36996570962543065875826224063, −6.23122845552399193509569567074, −5.53099012690230163518611277634, −5.18868553752006660434555978031, −3.33718247664778354944056942002, −2.13408356122024047575905306743, −1.30760507613651658153104125518,
0.866246506359621768826081250832, 1.61675823798770280219010353059, 2.44287902446317438221797576635, 3.97662670000536791222486889627, 4.35727221415778055157997046292, 6.39883572281414926613117854844, 7.09183456863362862272954223706, 7.26208451296680981269798434185, 8.074302514414984218025237682282, 8.816482317921814072441864105489