| L(s) = 1 | + 352. i·3-s − 3.77e3·5-s − 1.56e4i·7-s − 6.49e4·9-s + 1.15e5i·11-s + 5.46e5·13-s − 1.32e6i·15-s − 1.50e6·17-s + 3.45e6i·19-s + 5.49e6·21-s + 3.90e6i·23-s + 4.47e6·25-s − 2.06e6i·27-s − 1.57e7·29-s + 1.36e7i·31-s + ⋯ |
| L(s) = 1 | + 1.44i·3-s − 1.20·5-s − 0.929i·7-s − 1.09·9-s + 0.715i·11-s + 1.47·13-s − 1.74i·15-s − 1.05·17-s + 1.39i·19-s + 1.34·21-s + 0.607i·23-s + 0.458·25-s − 0.144i·27-s − 0.767·29-s + 0.476i·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
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.2763707702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2763707702\) |
| \(L(6)\) |
|
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 | \( 1 - 352. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 3.77e3T + 9.76e6T^{2} \) |
| 7 | \( 1 + 1.56e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 1.15e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 5.46e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 1.50e6T + 2.01e12T^{2} \) |
| 19 | \( 1 - 3.45e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 3.90e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 1.57e7T + 4.20e14T^{2} \) |
| 31 | \( 1 - 1.36e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 7.96e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 1.01e7T + 1.34e16T^{2} \) |
| 43 | \( 1 + 4.09e7iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 2.36e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 3.09e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 6.26e7iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 1.02e9T + 7.13e17T^{2} \) |
| 67 | \( 1 - 6.25e8iT - 1.82e18T^{2} \) |
| 71 | \( 1 - 2.05e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 2.44e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 3.04e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 2.75e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 2.51e8T + 3.11e19T^{2} \) |
| 97 | \( 1 + 1.56e10T + 7.37e19T^{2} \) |
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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
−10.93196723562030883156263434501, −10.12884734697639363479217012964, −9.158215580886606313354401729987, −8.150952874088808765987719682040, −7.25178260260221918575608743313, −5.89088417194186990505849165478, −4.43081441744016231910201354393, −4.06356314071163213305952427361, −3.32966614621039994372993873170, −1.39186111795923367554638599294,
0.06992137954037548468420830535, 0.820755056606584775498429531149, 2.07619506616811580338744048504, 3.10170105796457363034693858925, 4.36031369076305403700785542429, 5.89882551368782298822443482931, 6.60582212184768765016359483499, 7.62479411891386249633368079180, 8.452780020288846801095593173395, 9.006912053410853832482244192084
