| L(s) = 1 | − i·2-s − i·3-s + 4-s − 6-s − 4i·7-s − 3i·8-s − 9-s + 11-s − i·12-s − 2i·13-s − 4·14-s − 16-s + 2i·17-s + i·18-s − 4·21-s − i·22-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s + 0.5·4-s − 0.408·6-s − 1.51i·7-s − 1.06i·8-s − 0.333·9-s + 0.301·11-s − 0.288i·12-s − 0.554i·13-s − 1.06·14-s − 0.250·16-s + 0.485i·17-s + 0.235i·18-s − 0.872·21-s − 0.213i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.398818 - 1.68942i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.398818 - 1.68942i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−10.19827260779541492414597944516, −9.216227997030175190811964392102, −7.902016678713036797508885615016, −7.28333173292208514000135672280, −6.62075853550757221784196120429, −5.50583698078919530729031551332, −3.98217339862346779260223284124, −3.33165441182062053373742244392, −1.88885768745628063733273141121, −0.856711915194970839414152836025,
2.12216472826106110222686091145, 3.01455761281709906487139113814, 4.58884225367745366365122857165, 5.41211465666534224643436194700, 6.25940605281211111235736647426, 6.92888766118018441161434480736, 8.234062251810906359382240114906, 8.731712903125441223769712013187, 9.555443656409105116442124256140, 10.59894708919633338728836753681
