| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 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
−9.539561210487403574675283344717, −8.720780437434469962417611550224, −7.947734946671294662176519651630, −6.76688402776421694590616068181, −5.89685145504248507853114839610, −5.00990142224555417387955616255, −4.08751456399959515893395846470, −3.48532229815160737435781499688, −2.05326687888168253012791912667, 0,
1.71445765527751321897892714700, 2.61865819129094450462582354274, 3.79555062207260394056854598536, 4.92788450814285464970946220317, 6.38643347295196248226107769990, 6.55439813991020394176486152580, 7.33782374831349734915177782979, 8.230129938830794240392666925692, 9.157322000026837505816158343106
