| L(s) = 1 | − 121. i·3-s − 2.68e3·5-s + 3.02e4i·7-s + 4.42e4·9-s + 3.31e4i·11-s − 5.10e5·13-s + 3.27e5i·15-s − 2.64e6·17-s − 3.77e6i·19-s + 3.68e6·21-s − 6.21e6i·23-s − 2.53e6·25-s − 1.25e7i·27-s + 3.22e7·29-s + 1.24e7i·31-s + ⋯ |
| L(s) = 1 | − 0.500i·3-s − 0.860·5-s + 1.80i·7-s + 0.749·9-s + 0.205i·11-s − 1.37·13-s + 0.430i·15-s − 1.86·17-s − 1.52i·19-s + 0.901·21-s − 0.966i·23-s − 0.259·25-s − 0.875i·27-s + 1.57·29-s + 0.436i·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
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.067368933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067368933\) |
| \(L(6)\) |
|
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 | \( 1 + 121. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 2.68e3T + 9.76e6T^{2} \) |
| 7 | \( 1 - 3.02e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 3.31e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 5.10e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 2.64e6T + 2.01e12T^{2} \) |
| 19 | \( 1 + 3.77e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 6.21e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 3.22e7T + 4.20e14T^{2} \) |
| 31 | \( 1 - 1.24e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + 6.65e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + 9.97e6T + 1.34e16T^{2} \) |
| 43 | \( 1 + 2.31e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 + 4.86e7iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 1.33e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 8.52e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 4.10e8T + 7.13e17T^{2} \) |
| 67 | \( 1 - 1.27e9iT - 1.82e18T^{2} \) |
| 71 | \( 1 - 1.06e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 3.60e8T + 4.29e18T^{2} \) |
| 79 | \( 1 - 2.41e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 2.05e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 6.58e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 1.54e9T + 7.37e19T^{2} \) |
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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
−10.23346748678158954019975218391, −9.000360977382895468782626882563, −8.476316072956508565992536194019, −7.15781486062538381791338301123, −6.60809486464495152777236847640, −5.08676175583002466280785035581, −4.37950769478567145917622059082, −2.60410409989816753200682448184, −2.16707325241761906657590958376, −0.43002371051675145998083168239,
0.44713114050546916880578092637, 1.70367377765677451221630659932, 3.41510790772822219684175596866, 4.23145709610012655674213116871, 4.74841919342953070924098393274, 6.59748633120438827044829943727, 7.39579824282011493787734326763, 8.073219040141072518718455849205, 9.580963031570972605230617090843, 10.25202278110767029723441490576
