| L(s) = 1 | + (−8.14 + 7.84i)2-s + (4.82 − 127. i)4-s + 294.·5-s + 328. i·7-s + (964. + 1.08e3i)8-s + (−2.39e3 + 2.30e3i)10-s − 2.68e3i·11-s + 1.48e3i·13-s + (−2.57e3 − 2.67e3i)14-s + (−1.63e4 − 1.23e3i)16-s − 3.55e4i·17-s + 1.60e4·19-s + (1.42e3 − 3.76e4i)20-s + (2.10e4 + 2.18e4i)22-s − 1.15e4·23-s + ⋯ |
| L(s) = 1 | + (−0.720 + 0.693i)2-s + (0.0377 − 0.999i)4-s + 1.05·5-s + 0.362i·7-s + (0.665 + 0.745i)8-s + (−0.758 + 0.730i)10-s − 0.607i·11-s + 0.187i·13-s + (−0.251 − 0.260i)14-s + (−0.997 − 0.0753i)16-s − 1.75i·17-s + 0.538·19-s + (0.0397 − 1.05i)20-s + (0.421 + 0.437i)22-s − 0.198·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.58255 + 0.0897744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58255 + 0.0897744i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (8.14 - 7.84i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 294.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 328. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 2.68e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 1.48e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 3.55e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 1.60e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.15e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.04e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.49e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 6.15e4iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 2.32e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 6.19e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.07e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.13e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.30e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.52e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 4.81e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.36e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.23e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.34e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 6.96e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 1.66e5iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 7.05e6T + 8.07e13T^{2} \) |
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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
−13.83584308643694512595138133258, −11.96890262331175730519273166740, −10.63280395343511867093284889921, −9.558259778636612859174696827082, −8.783036482430322476521166022332, −7.29494579591622249705942755764, −6.07746124856614805835767862955, −5.08940784444543945223770874746, −2.47363488733064878692235500241, −0.840466609008814792193393130098,
1.17385728142469410747538352719, 2.42063986171513241744842321600, 4.14963193981860739226357921912, 6.05907410072252819633428377591, 7.53002368952040096487064685143, 8.813241904268360702784337971968, 10.00062221418018442699545757283, 10.54973126220691421995758061113, 12.03636465361645132274889625309, 13.02326711438629071565849943625
