| L(s) = 1 | + 357. i·3-s + 1.63e3·5-s + 3.37e3i·7-s − 6.85e4·9-s − 2.21e5i·11-s − 3.74e5·13-s + 5.82e5i·15-s − 4.30e5·17-s + 3.03e5i·19-s − 1.20e6·21-s + 8.29e6i·23-s − 7.10e6·25-s − 3.38e6i·27-s − 3.72e7·29-s − 7.54e6i·31-s + ⋯ |
| L(s) = 1 | + 1.46i·3-s + 0.521·5-s + 0.201i·7-s − 1.16·9-s − 1.37i·11-s − 1.00·13-s + 0.767i·15-s − 0.303·17-s + 0.122i·19-s − 0.295·21-s + 1.28i·23-s − 0.727·25-s − 0.236i·27-s − 1.81·29-s − 0.263i·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
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.475383840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.475383840\) |
| \(L(6)\) |
|
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 | \( 1 - 357. iT - 5.90e4T^{2} \) |
| 5 | \( 1 - 1.63e3T + 9.76e6T^{2} \) |
| 7 | \( 1 - 3.37e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 2.21e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 3.74e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 4.30e5T + 2.01e12T^{2} \) |
| 19 | \( 1 - 3.03e5iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 8.29e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 3.72e7T + 4.20e14T^{2} \) |
| 31 | \( 1 + 7.54e6iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 7.50e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + 6.61e7T + 1.34e16T^{2} \) |
| 43 | \( 1 + 2.22e7iT - 2.16e16T^{2} \) |
| 47 | \( 1 + 1.88e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 5.80e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 3.07e7iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 5.40e8T + 7.13e17T^{2} \) |
| 67 | \( 1 - 2.14e9iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 2.89e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 3.51e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 4.62e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + 2.00e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 5.96e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 5.56e9T + 7.37e19T^{2} \) |
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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
−10.06232175275288214552235611670, −9.472018292791532161573913360605, −8.678979963780994580025736737321, −7.41140370207201901005397216859, −5.81367888681744948396843905961, −5.36154161725367810464828823727, −4.09209159685366563237427324073, −3.26714570884529310370574970156, −2.03673881334076742998662655683, −0.33600865315850883878087968623,
0.790603899515845029358129925040, 2.01405367293294878281737716561, 2.38517494236811279668731051577, 4.24663664843466620116888477239, 5.44997393870852080305686973110, 6.60940323625769389450392285698, 7.22965715217645803315306676691, 7.999921367017748905043289336968, 9.328280127048232170817917348738, 10.12993066392640800764587038470
