| 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 | \( 1 - 1.47e8T + 7.83e15T^{2} \) |
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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
−25.48457455056030862586971118301, −22.61633771030336282354266566175, −21.40363457458476111200117821823, −20.12216169702305363155548438682, −18.82264965358014825023913936402, −15.90659526368572752200092385994, −13.54027667979888221166920257283, −11.13538667100853805576196445480, −9.509896719961398489491287140507, −3.29129434150346804494000997655,
6.56108305736119635078666434601, 8.537722471016272724437647432597, 12.97742864252240100208929908279, 14.76513444245845090232475090568, 16.52349202435110678096070736482, 18.29684445410031745836940228720, 20.23187960048157825678945100376, 23.03125818036933381721165597225, 24.24988930999681135601755894035, 25.43404712791207914444593543883
