| L(s) = 1 | − 39.9i·3-s − 279.·5-s + 2.63e3i·7-s + 4.96e3·9-s + 2.48e3i·11-s − 4.13e4·13-s + 1.11e4i·15-s − 4.86e4·17-s − 2.41e5i·19-s + 1.05e5·21-s + 5.77e4i·23-s + 7.81e4·25-s − 4.60e5i·27-s + 7.29e5·29-s + 1.34e5i·31-s + ⋯ |
| L(s) = 1 | − 0.493i·3-s − 0.447·5-s + 1.09i·7-s + 0.756·9-s + 0.169i·11-s − 1.44·13-s + 0.220i·15-s − 0.582·17-s − 1.85i·19-s + 0.541·21-s + 0.206i·23-s + 0.200·25-s − 0.866i·27-s + 1.03·29-s + 0.146i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.609825493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609825493\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 279.T \) |
| good | 3 | \( 1 + 39.9iT - 6.56e3T^{2} \) |
| 7 | \( 1 - 2.63e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 2.48e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.13e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 4.86e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 2.41e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 5.77e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 7.29e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.34e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 1.68e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 7.61e5T + 7.98e12T^{2} \) |
| 43 | \( 1 - 2.54e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 5.65e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.14e7T + 6.22e13T^{2} \) |
| 59 | \( 1 - 2.12e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.63e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.53e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 2.06e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.51e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 1.95e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 2.37e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.86e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 9.46e7T + 7.83e15T^{2} \) |
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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
−10.16094651368253610275281564265, −9.245642726571706982045199836008, −8.369954013821059205187321308676, −7.21574956827780320415189653703, −6.69597393762402199991311237881, −5.20488504873265484382791319977, −4.46125722745844851211477829169, −2.82427856455002818998936654881, −2.05458460165855255316308958355, −0.60023103892254110020782411565,
0.54197462719210102578812089203, 1.84098777931275019809027480953, 3.41350326282733103186608321715, 4.24573998579072494126257925287, 5.03051644924673803315331398421, 6.58477187083262833611332242567, 7.41848788223051536490447990226, 8.220670900964382214206803562331, 9.620143422891905039144558344977, 10.20429466715502135421327631378
