| 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
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(3.401204646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.401204646\) |
| \(L(6)\) |
|
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
−10.42384871466552966454721972524, −9.739289511157929324548300203820, −9.063502040374265813437129996657, −8.451598993881492755199869946786, −6.11230155039785265088205146935, −5.67740295226610670772525202358, −5.04201217912665592090763491269, −3.45594058543358069284980864890, −2.69025009614622544610790641338, −1.43003130183341686906081445767,
0.61502833785180800949470482663, 1.35657459804476110372260032091, 1.96886312905184921258635101804, 3.26159099045265710925391207403, 5.01952471846850284865741030208, 6.20201389939800891757470640737, 6.73286573446295002303183888234, 7.57955488780060274040636832755, 8.629359735629741441277638427379, 9.950728714949467660519130071746
