| L(s) = 1 | − 418. i·3-s − 5.46e3·5-s + 2.62e4i·7-s − 1.16e5·9-s + 1.10e5i·11-s − 3.33e5·13-s + 2.28e6i·15-s + 1.74e6·17-s − 5.79e5i·19-s + 1.10e7·21-s + 4.39e6i·23-s + 2.00e7·25-s + 2.39e7i·27-s − 1.17e7·29-s − 1.47e6i·31-s + ⋯ |
| L(s) = 1 | − 1.72i·3-s − 1.74·5-s + 1.56i·7-s − 1.97·9-s + 0.685i·11-s − 0.897·13-s + 3.01i·15-s + 1.23·17-s − 0.234i·19-s + 2.69·21-s + 0.682i·23-s + 2.05·25-s + 1.67i·27-s − 0.572·29-s − 0.0514i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.03107775292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03107775292\) |
| \(L(6)\) |
|
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 | \( 1 + 418. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 5.46e3T + 9.76e6T^{2} \) |
| 7 | \( 1 - 2.62e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 1.10e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 3.33e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 1.74e6T + 2.01e12T^{2} \) |
| 19 | \( 1 + 5.79e5iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 4.39e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 1.17e7T + 4.20e14T^{2} \) |
| 31 | \( 1 + 1.47e6iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 4.17e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + 1.23e8T + 1.34e16T^{2} \) |
| 43 | \( 1 - 1.56e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 4.45e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 4.27e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + 9.43e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 3.87e8T + 7.13e17T^{2} \) |
| 67 | \( 1 - 4.47e8iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 6.97e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.66e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 5.43e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + 4.16e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 2.50e9T + 3.11e19T^{2} \) |
| 97 | \( 1 - 7.97e9T + 7.37e19T^{2} \) |
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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
−9.499367205537352822491205264921, −8.366040675912393547606151472145, −7.72056962183317130584116041181, −7.16481454388884201091201256440, −5.96769245550100407793156866167, −4.87392607447638215871915035363, −3.24508850356979936281565855449, −2.37365337527819478262949472310, −1.22262344546271727234208498290, −0.01008479798659954817646492061,
0.65494910483716344052201278988, 3.17293967969886064509527107638, 3.78785412064095224593225499918, 4.36199752549530348691263239966, 5.32640597840254253833203781208, 7.10151620374870430056091015482, 7.901237567773536721912775490471, 8.814975754289943668944539983934, 10.16865447079058802401626134127, 10.48076582622485025670796666372
