| L(s) = 1 | − 0.846i·2-s − 54.5i·3-s + 1.02e3·4-s − 4.88e3i·5-s − 46.1·6-s − 3.06e4i·7-s − 1.73e3i·8-s + 5.60e4·9-s − 4.13e3·10-s − 1.79e5·11-s − 5.58e4i·12-s − 3.38e5·13-s − 2.59e4·14-s − 2.66e5·15-s + 1.04e6·16-s + 7.66e5·17-s + ⋯ |
| L(s) = 1 | − 0.0264i·2-s − 0.224i·3-s + 0.999·4-s − 1.56i·5-s − 0.00593·6-s − 1.82i·7-s − 0.0528i·8-s + 0.949·9-s − 0.0413·10-s − 1.11·11-s − 0.224i·12-s − 0.911·13-s − 0.0481·14-s − 0.350·15-s + 0.997·16-s + 0.539·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.669353 - 2.10647i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.669353 - 2.10647i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.20e8 + 8.48e7i)T \) |
| good | 2 | \( 1 + 0.846iT - 1.02e3T^{2} \) |
| 3 | \( 1 + 54.5iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 4.88e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 + 3.06e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 1.79e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + 3.38e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 7.66e5T + 2.01e12T^{2} \) |
| 19 | \( 1 - 3.89e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 3.13e6T + 4.14e13T^{2} \) |
| 29 | \( 1 - 1.81e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 2.96e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + 8.05e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 - 4.52e7T + 1.34e16T^{2} \) |
| 47 | \( 1 - 1.36e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + 3.81e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 1.06e9T + 5.11e17T^{2} \) |
| 61 | \( 1 + 7.22e8iT - 7.13e17T^{2} \) |
| 67 | \( 1 - 1.44e9T + 1.82e18T^{2} \) |
| 71 | \( 1 + 3.79e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 1.13e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 4.58e9T + 9.46e18T^{2} \) |
| 83 | \( 1 - 6.19e9T + 1.55e19T^{2} \) |
| 89 | \( 1 + 7.46e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 3.79e9T + 7.37e19T^{2} \) |
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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
−12.87036057766167688312303336604, −12.49434474726854007399732400788, −10.71876792522165712005882030296, −9.871573875060642626578178346112, −7.83290768935709968683112689072, −7.24799276050581089478154236092, −5.30053178719435587551213756244, −3.91154106139746228186690903324, −1.68138611818319326287570076046, −0.68326096846780782324351477062,
2.35137859192915080407905624327, 2.84761396207407435378252137709, 5.35510915146534396074080038708, 6.68043996355485809839068603559, 7.67046943064288807532801437297, 9.620842160088433836985410614044, 10.71886198520683050972350481628, 11.67240170813278747671482681367, 12.84333375729468729637661207940, 14.79717474456796988988003695335
