| L(s) = 1 | + (−8.10 + 3.91i)3-s + 29.9·5-s + 5.14·7-s + (50.3 − 63.4i)9-s − 170.·11-s + 161. i·13-s + (−242. + 117. i)15-s − 32.6i·17-s − 299. i·19-s + (−41.6 + 20.1i)21-s + 333. i·23-s + 271.·25-s + (−159. + 711. i)27-s + 930.·29-s − 128.·31-s + ⋯ |
| L(s) = 1 | + (−0.900 + 0.434i)3-s + 1.19·5-s + 0.104·7-s + (0.621 − 0.783i)9-s − 1.40·11-s + 0.954i·13-s + (−1.07 + 0.520i)15-s − 0.112i·17-s − 0.829i·19-s + (−0.0945 + 0.0456i)21-s + 0.630i·23-s + 0.434·25-s + (−0.219 + 0.975i)27-s + 1.10·29-s − 0.133·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6314724458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6314724458\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (8.10 - 3.91i)T \) |
| good | 5 | \( 1 - 29.9T + 625T^{2} \) |
| 7 | \( 1 - 5.14T + 2.40e3T^{2} \) |
| 11 | \( 1 + 170.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 161. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 32.6iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 299. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 333. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 930.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 128.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 2.41e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.19e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.96e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 3.95e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.58e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.88e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 1.61e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 8.28e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 3.15e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.56e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 9.86e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 8.99e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 1.59e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 2.54e3T + 8.85e7T^{2} \) |
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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
−11.07630572008267344412753003735, −10.07926642185177638549549799340, −9.686431168662006657643663658377, −8.510494383568630760397056112136, −7.08850911341542144088154393933, −6.21860651020239396471279463064, −5.28861173070491653773603907192, −4.58174675392316150563846145087, −2.84473902843777019458916671314, −1.46938689670836000382356157869,
0.19447249241383956119000768842, 1.63292631596548892345648973129, 2.76538210312239560616134365264, 4.72543381483496279222679554571, 5.64060034150546114898357266166, 6.15229368626085239666983635991, 7.45742842563494389039956873963, 8.259297377537949146304094699501, 9.710846160945261431100641309689, 10.46064969147547561359207749355
