| L(s) = 1 | − 18.7i·2-s − 59.7i·3-s − 222.·4-s − 1.11e3·6-s − 438. i·7-s + 1.76e3i·8-s − 1.38e3·9-s + 5.75e3·11-s + 1.32e4i·12-s − 3.53e3i·13-s − 8.20e3·14-s + 4.59e3·16-s + 2.39e4i·17-s + 2.58e4i·18-s − 1.65e4·19-s + ⋯ |
| L(s) = 1 | − 1.65i·2-s − 1.27i·3-s − 1.73·4-s − 2.11·6-s − 0.482i·7-s + 1.21i·8-s − 0.631·9-s + 1.30·11-s + 2.21i·12-s − 0.445i·13-s − 0.798·14-s + 0.280·16-s + 1.18i·17-s + 1.04i·18-s − 0.554·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.728207 + 1.17826i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.728207 + 1.17826i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 18.7iT - 128T^{2} \) |
| 3 | \( 1 + 59.7iT - 2.18e3T^{2} \) |
| 7 | \( 1 + 438. iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 5.75e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 3.53e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.39e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 1.65e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.56e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.34e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.61e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 3.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.88e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 3.43e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 1.66e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 2.54e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.52e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.56e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 2.99e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.12e5iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 1.95e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.21e5iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 5.78e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.20e6iT - 8.07e13T^{2} \) |
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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
−14.52712154078789224781585868152, −13.23305867846717043056502988855, −12.53577384234591590726570937912, −11.45339351758365206346347323225, −10.14944644776911907610308109601, −8.479424157938103499013994734579, −6.63339446356861065717915009979, −3.92741258083991187130266315639, −1.98009051029613126538203809654, −0.77233587109972640340389006857,
4.11578602931610703805526971830, 5.44491928207872422184003058857, 6.97066732898779448653583551237, 8.814845782304262610268102430148, 9.611315210265091139137820919374, 11.59660192020129146121666336159, 13.78132683857699356426219474089, 14.86032092982336772290800983151, 15.60382318798963268141776095919, 16.55262584153428437480523642037
