| L(s) = 1 | − i·3-s − i·5-s − 9-s + 4i·11-s − 2i·13-s − 15-s − 2·17-s − 8i·19-s − 4·23-s − 25-s + i·27-s + 6i·29-s + 4·33-s + 2i·37-s − 2·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.447i·5-s − 0.333·9-s + 1.20i·11-s − 0.554i·13-s − 0.258·15-s − 0.485·17-s − 1.83i·19-s − 0.834·23-s − 0.200·25-s + 0.192i·27-s + 1.11i·29-s + 0.696·33-s + 0.328i·37-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + 14iT - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.075263460359479750511363664537, −7.14738697888463370612283972642, −6.79167009903293050975870440860, −5.80949005153059272425789435923, −4.90191953837499255577179687411, −4.43293941909340970940938807943, −3.15224091191776120944315956507, −2.28082523926521492288051502464, −1.32706305741639967784361735240, 0,
1.66091355616582563257249936183, 2.74001181536765894131453375515, 3.69499445721257335797393812289, 4.15276760460354368331489310057, 5.28245199162719436119870350016, 6.10319062723245459059555705664, 6.43615873796510427765755664625, 7.72973860925260609962130087477, 8.142655882146513320848952342960
