| L(s) = 1 | + 0.426i·3-s − 59.7i·7-s + 242.·9-s − 121·11-s + 1.12e3i·13-s + 1.94e3i·17-s + 1.24e3·19-s + 25.5·21-s + 4.64e3i·23-s + 207. i·27-s − 1.00e3·29-s − 2.47e3·31-s − 51.6i·33-s − 1.08e4i·37-s − 480.·39-s + ⋯ |
| L(s) = 1 | + 0.0273i·3-s − 0.461i·7-s + 0.999·9-s − 0.301·11-s + 1.84i·13-s + 1.63i·17-s + 0.791·19-s + 0.0126·21-s + 1.82i·23-s + 0.0547i·27-s − 0.220·29-s − 0.463·31-s − 0.00825i·33-s − 1.30i·37-s − 0.0505·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.279084123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.279084123\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 - 0.426iT - 243T^{2} \) |
| 7 | \( 1 + 59.7iT - 1.68e4T^{2} \) |
| 13 | \( 1 - 1.12e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.94e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.24e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.64e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.08e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.17e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.90e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 4.95e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 8.17e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.69e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 825. iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.19e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.90e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 2.63e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.08e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 3.21e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.65e5iT - 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.498560079866016655505892392567, −8.791865829040937068101513601527, −7.57754385064486987523433848733, −7.15933733385467007688169452934, −6.19187474705967285284262471136, −5.17732855128147697143688378791, −4.07703658589769479182515312991, −3.64194948117482996007648771563, −1.90963578624285681922389540821, −1.38745606187390969933632660655,
0.24146969983669057302087796865, 1.15789956436578743935704241579, 2.58220651382054327378960409091, 3.22412698211312693734720984964, 4.65559650281075897363165264012, 5.21932350151712180157554188815, 6.24445235678337608286356794733, 7.24790683299201263957619928648, 7.85971102231806112246870598109, 8.762323890089930716166785355517
