L(s) = 1 | − 2-s − 31·4-s + 34·5-s − 49·7-s + 63·8-s − 34·10-s + 340·11-s + 454·13-s + 49·14-s + 929·16-s + 798·17-s + 892·19-s − 1.05e3·20-s − 340·22-s + 3.19e3·23-s − 1.96e3·25-s − 454·26-s + 1.51e3·28-s + 8.24e3·29-s − 2.49e3·31-s − 2.94e3·32-s − 798·34-s − 1.66e3·35-s + 9.79e3·37-s − 892·38-s + 2.14e3·40-s − 1.98e4·41-s + ⋯ |
L(s) = 1 | − 0.176·2-s − 0.968·4-s + 0.608·5-s − 0.377·7-s + 0.348·8-s − 0.107·10-s + 0.847·11-s + 0.745·13-s + 0.0668·14-s + 0.907·16-s + 0.669·17-s + 0.566·19-s − 0.589·20-s − 0.149·22-s + 1.25·23-s − 0.630·25-s − 0.131·26-s + 0.366·28-s + 1.81·29-s − 0.466·31-s − 0.508·32-s − 0.118·34-s − 0.229·35-s + 1.17·37-s − 0.100·38-s + 0.211·40-s − 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.460275632\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.460275632\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
good | 2 | \( 1 + T + p^{5} T^{2} \) |
| 5 | \( 1 - 34 T + p^{5} T^{2} \) |
| 11 | \( 1 - 340 T + p^{5} T^{2} \) |
| 13 | \( 1 - 454 T + p^{5} T^{2} \) |
| 17 | \( 1 - 798 T + p^{5} T^{2} \) |
| 19 | \( 1 - 892 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3192 T + p^{5} T^{2} \) |
| 29 | \( 1 - 8242 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2496 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9798 T + p^{5} T^{2} \) |
| 41 | \( 1 + 19834 T + p^{5} T^{2} \) |
| 43 | \( 1 + 17236 T + p^{5} T^{2} \) |
| 47 | \( 1 + 8928 T + p^{5} T^{2} \) |
| 53 | \( 1 + 150 T + p^{5} T^{2} \) |
| 59 | \( 1 - 42396 T + p^{5} T^{2} \) |
| 61 | \( 1 - 14758 T + p^{5} T^{2} \) |
| 67 | \( 1 + 1676 T + p^{5} T^{2} \) |
| 71 | \( 1 + 14568 T + p^{5} T^{2} \) |
| 73 | \( 1 - 78378 T + p^{5} T^{2} \) |
| 79 | \( 1 + 2272 T + p^{5} T^{2} \) |
| 83 | \( 1 - 37764 T + p^{5} T^{2} \) |
| 89 | \( 1 - 117286 T + p^{5} T^{2} \) |
| 97 | \( 1 - 10002 T + p^{5} T^{2} \) |
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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
−13.82146605035704294422884079869, −13.11966601210301264113418151012, −11.78600831121606921028534122277, −10.20104993110059503481552408139, −9.371102911638747706506104033573, −8.303963228634616533879968982730, −6.55573956670720736585811620798, −5.15085999768495658742897163097, −3.52119467218360483878114964823, −1.09741881827720846859038886431,
1.09741881827720846859038886431, 3.52119467218360483878114964823, 5.15085999768495658742897163097, 6.55573956670720736585811620798, 8.303963228634616533879968982730, 9.371102911638747706506104033573, 10.20104993110059503481552408139, 11.78600831121606921028534122277, 13.11966601210301264113418151012, 13.82146605035704294422884079869