| L(s) = 1 | + 119. i·3-s − 948.·5-s − 1.54e4i·7-s + 4.48e4·9-s − 1.66e5i·11-s − 3.49e5·13-s − 1.13e5i·15-s − 1.33e6·17-s + 1.84e6i·19-s + 1.84e6·21-s − 9.06e6i·23-s − 8.86e6·25-s + 1.23e7i·27-s − 3.86e7·29-s + 3.73e7i·31-s + ⋯ |
| L(s) = 1 | + 0.490i·3-s − 0.303·5-s − 0.919i·7-s + 0.759·9-s − 1.03i·11-s − 0.940·13-s − 0.149i·15-s − 0.943·17-s + 0.743i·19-s + 0.451·21-s − 1.40i·23-s − 0.907·25-s + 0.863i·27-s − 1.88·29-s + 1.30i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.8986694154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8986694154\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 119. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 948.T + 9.76e6T^{2} \) |
| 7 | \( 1 + 1.54e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 1.66e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 3.49e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 1.33e6T + 2.01e12T^{2} \) |
| 19 | \( 1 - 1.84e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 9.06e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 3.86e7T + 4.20e14T^{2} \) |
| 31 | \( 1 - 3.73e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 2.87e6T + 4.80e15T^{2} \) |
| 41 | \( 1 - 1.14e8T + 1.34e16T^{2} \) |
| 43 | \( 1 + 1.12e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 + 2.25e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 3.16e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 4.46e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 1.42e9T + 7.13e17T^{2} \) |
| 67 | \( 1 - 7.31e7iT - 1.82e18T^{2} \) |
| 71 | \( 1 - 6.70e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 1.85e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 3.97e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 3.39e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 2.22e9T + 3.11e19T^{2} \) |
| 97 | \( 1 - 1.09e10T + 7.37e19T^{2} \) |
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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
−10.50966935509581244244725307578, −9.672302883341335834381261538574, −8.590860422700707535916693911717, −7.52102032430604513529402915712, −6.72210887431053434754916796464, −5.36931711250745758569731303578, −4.22980811278328715266442532660, −3.64609313924952889879509127904, −2.14267490512859284612921566624, −0.76848928454010586315751605192,
0.21494375195491742724002121359, 1.79417046971522039465821905707, 2.38225768751226120887704458710, 3.98607489527828900952320235286, 4.95530108613723971016841416892, 6.12148788942985232763505551882, 7.32126297449059033791863656794, 7.72808735476097588715352531450, 9.328524919634455987917325526330, 9.652941200635478703521509600288
