L(s) = 1 | + 40.3i·3-s − 186.·5-s − 202.·7-s − 898.·9-s + 1.59e3·11-s − 143. i·13-s − 7.53e3i·15-s + 4.77e3·17-s + (1.01e3 − 6.78e3i)19-s − 8.16e3i·21-s − 7.73e3·23-s + 1.92e4·25-s − 6.83e3i·27-s − 8.82e3i·29-s − 5.15e4i·31-s + ⋯ |
L(s) = 1 | + 1.49i·3-s − 1.49·5-s − 0.589·7-s − 1.23·9-s + 1.19·11-s − 0.0652i·13-s − 2.23i·15-s + 0.971·17-s + (0.147 − 0.989i)19-s − 0.881i·21-s − 0.635·23-s + 1.23·25-s − 0.347i·27-s − 0.361i·29-s − 1.73i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.304776465\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.304776465\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-1.01e3 + 6.78e3i)T \) |
good | 3 | \( 1 - 40.3iT - 729T^{2} \) |
| 5 | \( 1 + 186.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 202.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.59e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 143. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 4.77e3T + 2.41e7T^{2} \) |
| 23 | \( 1 + 7.73e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 8.82e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 5.15e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 9.46e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 7.87e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.29e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 9.72e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 4.61e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 6.17e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 8.56e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.90e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 4.08e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.30e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 1.06e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 1.22e4T + 3.26e11T^{2} \) |
| 89 | \( 1 - 5.61e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.51e5iT - 8.32e11T^{2} \) |
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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
−10.94052049105469444889806475368, −9.851559672970981862178638030963, −9.231093024254099323618615635715, −8.192544681651200738541040002772, −7.12446836484124923138246840783, −5.80692736231653774026279808111, −4.40325497663230462702336220726, −3.94137889401728045138329344196, −3.02901169125959871083188973386, −0.63679406539094579183016351897,
0.58366868157076524199053447920, 1.55007785210807606773144178328, 3.19621504426441021176783101625, 4.07911109695345531751842579822, 5.85197663541574680817081868705, 6.81496108906776290714995400028, 7.53604322455283221658748497330, 8.219968405456893006350780046501, 9.310779469323519875472317621619, 10.73015394109102304242311416110