| L(s) = 1 | − 19.2i·2-s − 46.7·3-s − 115.·4-s + 525.·5-s + 901. i·6-s + 658.·7-s − 2.70e3i·8-s + 2.18e3·9-s − 1.01e4i·10-s + 1.45e4i·11-s + 5.41e3·12-s + 4.94e4i·13-s − 1.26e4i·14-s − 2.45e4·15-s − 8.17e4·16-s − 7.07e4·17-s + ⋯ |
| L(s) = 1 | − 1.20i·2-s − 0.577·3-s − 0.452·4-s + 0.840·5-s + 0.695i·6-s + 0.274·7-s − 0.660i·8-s + 0.333·9-s − 1.01i·10-s + 0.993i·11-s + 0.261·12-s + 1.72i·13-s − 0.330i·14-s − 0.485·15-s − 1.24·16-s − 0.847·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.787i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.615 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.5148872568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5148872568\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 46.7T \) |
| 59 | \( 1 + (7.46e6 + 9.54e6i)T \) |
| good | 2 | \( 1 + 19.2iT - 256T^{2} \) |
| 5 | \( 1 - 525.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 658.T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.45e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 4.94e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 7.07e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 9.48e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 2.32e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 7.42e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 6.44e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 2.71e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 5.07e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 4.10e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 1.20e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.48e6T + 6.22e13T^{2} \) |
| 61 | \( 1 - 3.27e6iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 1.38e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 1.94e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 3.54e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 4.27e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 4.18e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 1.03e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 9.94e7iT - 7.83e15T^{2} \) |
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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
−10.71451213222682747987739489340, −9.719735581054421529463169088071, −9.153087910780531970118508759486, −7.19659421003219019957478296190, −6.36447350381837151623467881317, −4.89638771696372674051960441316, −3.90972095935436681014019221141, −2.05947947641011359776506367823, −1.78616067249876820948775238300, −0.11922647098181544272201726352,
1.44042420377395644714565828235, 3.07568793288905159042497880746, 5.06463975113135297439967715074, 5.66194945261372703570164007686, 6.45186118705304622914214976885, 7.64209355077122918744145332102, 8.499435265091143670184376960010, 9.775229559605538507964960733071, 10.86830082741459062270341758682, 11.67257354584741076526399302315
