| L(s) = 1 | + 3.93i·5-s − 1.11i·7-s − 3.26·11-s − 0.249·13-s + 5.86i·17-s − 2.19i·19-s − 5.58·23-s − 10.4·25-s − 2.72i·29-s − 10.3i·31-s + 4.36·35-s − 0.333·37-s − 6.10i·41-s + 9.82i·43-s − 9.41·47-s + ⋯ |
| L(s) = 1 | + 1.76i·5-s − 0.419i·7-s − 0.984·11-s − 0.0692·13-s + 1.42i·17-s − 0.504i·19-s − 1.16·23-s − 2.09·25-s − 0.505i·29-s − 1.86i·31-s + 0.738·35-s − 0.0547·37-s − 0.952i·41-s + 1.49i·43-s − 1.37·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1104742163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1104742163\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.93iT - 5T^{2} \) |
| 7 | \( 1 + 1.11iT - 7T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 + 0.249T + 13T^{2} \) |
| 17 | \( 1 - 5.86iT - 17T^{2} \) |
| 19 | \( 1 + 2.19iT - 19T^{2} \) |
| 23 | \( 1 + 5.58T + 23T^{2} \) |
| 29 | \( 1 + 2.72iT - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 0.333T + 37T^{2} \) |
| 41 | \( 1 + 6.10iT - 41T^{2} \) |
| 43 | \( 1 - 9.82iT - 43T^{2} \) |
| 47 | \( 1 + 9.41T + 47T^{2} \) |
| 53 | \( 1 - 4.75iT - 53T^{2} \) |
| 59 | \( 1 - 6.53T + 59T^{2} \) |
| 61 | \( 1 + 2.14T + 61T^{2} \) |
| 67 | \( 1 - 0.578iT - 67T^{2} \) |
| 71 | \( 1 + 3.26T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 8.85iT - 79T^{2} \) |
| 83 | \( 1 - 4.68T + 83T^{2} \) |
| 89 | \( 1 + 4.75iT - 89T^{2} \) |
| 97 | \( 1 + 1.83T + 97T^{2} \) |
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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
−9.649252178045552058457261011992, −8.340538255007829314081600225127, −7.71146250836423046984768441610, −7.17876295470069703603327690473, −6.16886111514152691823009691956, −5.87241115616213599155557039898, −4.42071999077856737948625636443, −3.67831161273209638968948737879, −2.74989171441650088549553574242, −2.00062084934081637070422989851,
0.03529180182747221178303250144, 1.31419158217756353107592882100, 2.40103790872485385082289188768, 3.58422344866297063991434967016, 4.74192642079702990195513594923, 5.14734864493555986974350643683, 5.78054458469491451768851461777, 6.96648436124550305004061153875, 7.87755231137466009407519470045, 8.471657466765163237650420841210
