| L(s) = 1 | − 2i·3-s − i·5-s + 2·7-s − 9-s − 2i·13-s − 2·15-s − 6·17-s − 4i·19-s − 4i·21-s + 6·23-s − 25-s − 4i·27-s − 6i·29-s + 4·31-s − 2i·35-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s − 0.447i·5-s + 0.755·7-s − 0.333·9-s − 0.554i·13-s − 0.516·15-s − 1.45·17-s − 0.917i·19-s − 0.872i·21-s + 1.25·23-s − 0.200·25-s − 0.769i·27-s − 1.11i·29-s + 0.718·31-s − 0.338i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.608077633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.608077633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.148085412675813891936646785210, −8.385500135164812514756384431203, −7.78517028695477112170101699191, −6.89851601602176498797041632823, −6.27477430921408277322119933101, −5.04171054891648705236366295193, −4.42029860877984215347666245476, −2.81019808355718984241074660766, −1.81280180084146899617098350177, −0.68873071227057691255821670662,
1.71448922623350348090211524349, 3.03500349830528548119990947945, 4.11355726121156523173933348244, 4.69285146601263802022539723429, 5.59836186515787395494516653704, 6.76048083043038491165210125637, 7.44710099286074170155524085338, 8.801834284339849304294155559021, 8.959129842932469534164806714545, 10.20052898539242611397909731122
