| L(s) = 1 | − 10.8i·3-s + 163.·7-s + 125.·9-s + 321. i·11-s + 128. i·13-s + 2.11e3·17-s − 1.45e3i·19-s − 1.77e3i·21-s − 1.23e3·23-s − 3.99e3i·27-s − 4.07e3i·29-s + 3.95e3·31-s + 3.48e3·33-s + 1.06e4i·37-s + 1.38e3·39-s + ⋯ |
| L(s) = 1 | − 0.694i·3-s + 1.26·7-s + 0.517·9-s + 0.801i·11-s + 0.210i·13-s + 1.77·17-s − 0.924i·19-s − 0.876i·21-s − 0.485·23-s − 1.05i·27-s − 0.899i·29-s + 0.739·31-s + 0.556·33-s + 1.27i·37-s + 0.146·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.401i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.916 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.276070554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.276070554\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 10.8iT - 243T^{2} \) |
| 7 | \( 1 - 163.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 321. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 128. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 2.11e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.45e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.07e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 3.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.06e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 5.90e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.64e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.32e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.06e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.52e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.91e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.08e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.25e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.68e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 9.04e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.49e5T + 8.58e9T^{2} \) |
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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
−9.585189552469009235325227743378, −8.287628451166840968394585625928, −7.74533402276687495466015300161, −7.06732554340907879891989474873, −6.01981283687711223955347470201, −4.90412359069932237208142370711, −4.22823348695818886718371934709, −2.69114412503748785940609290897, −1.62023291241129660448397992529, −0.943379521777699350879158837354,
0.861632574432560896443446796816, 1.82297973055510487792006201151, 3.35732709908559186018502231512, 4.08633142180622286205486359532, 5.21485844081377832800335016336, 5.70530996476060679249241708415, 7.16456454288364769138357897848, 7.986923941088818240954468357385, 8.624613288568801105449537514887, 9.739353086142610263781823420249
