L(s) = 1 | + 22.4i·2-s + (45 − 67.3i)3-s − 248·4-s − 224. i·5-s + (1.51e3 + 1.01e3i)6-s − 1.75e3·7-s + 179. i·8-s + (−2.51e3 − 6.06e3i)9-s + 5.04e3·10-s + 6.95e3i·11-s + (−1.11e4 + 1.67e4i)12-s + 2.57e4·13-s − 3.92e4i·14-s + (−1.51e4 − 1.01e4i)15-s − 6.75e4·16-s + 7.48e4i·17-s + ⋯ |
L(s) = 1 | + 1.40i·2-s + (0.555 − 0.831i)3-s − 0.968·4-s − 0.359i·5-s + (1.16 + 0.779i)6-s − 0.728·7-s + 0.0438i·8-s + (−0.382 − 0.923i)9-s + 0.504·10-s + 0.475i·11-s + (−0.538 + 0.805i)12-s + 0.900·13-s − 1.02i·14-s + (−0.298 − 0.199i)15-s − 1.03·16-s + 0.896i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.03711 + 0.554359i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03711 + 0.554359i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-45 + 67.3i)T \) |
good | 2 | \( 1 - 22.4iT - 256T^{2} \) |
| 5 | \( 1 + 224. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.75e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 6.95e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.57e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 7.48e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.89e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 4.70e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 4.60e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 3.51e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.33e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.87e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 3.52e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 4.08e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 6.60e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.37e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 7.53e5T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.26e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.70e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.76e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 2.29e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 4.63e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 7.26e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.47e8T + 7.83e15T^{2} \) |
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
−25.43404712791207914444593543883, −24.24988930999681135601755894035, −23.03125818036933381721165597225, −20.23187960048157825678945100376, −18.29684445410031745836940228720, −16.52349202435110678096070736482, −14.76513444245845090232475090568, −12.97742864252240100208929908279, −8.537722471016272724437647432597, −6.56108305736119635078666434601,
3.29129434150346804494000997655, 9.509896719961398489491287140507, 11.13538667100853805576196445480, 13.54027667979888221166920257283, 15.90659526368572752200092385994, 18.82264965358014825023913936402, 20.12216169702305363155548438682, 21.40363457458476111200117821823, 22.61633771030336282354266566175, 25.48457455056030862586971118301