| L(s) = 1 | − 0.414i·3-s + 3.82i·5-s + 1.58·7-s + 2.82·9-s − 4.82i·11-s − i·13-s + 1.58·15-s + 17-s + 5.65i·19-s − 0.656i·21-s + 3.17·23-s − 9.65·25-s − 2.41i·27-s + 7.65i·29-s + 7.65·31-s + ⋯ |
| L(s) = 1 | − 0.239i·3-s + 1.71i·5-s + 0.599·7-s + 0.942·9-s − 1.45i·11-s − 0.277i·13-s + 0.409·15-s + 0.242·17-s + 1.29i·19-s − 0.143i·21-s + 0.661·23-s − 1.93·25-s − 0.464i·27-s + 1.42i·29-s + 1.37·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64449 + 0.681170i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64449 + 0.681170i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 3 | \( 1 + 0.414iT - 3T^{2} \) |
| 5 | \( 1 - 3.82iT - 5T^{2} \) |
| 7 | \( 1 - 1.58T + 7T^{2} \) |
| 11 | \( 1 + 4.82iT - 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 5.65iT - 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 - 7.65iT - 29T^{2} \) |
| 31 | \( 1 - 7.65T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 - 5.58iT - 43T^{2} \) |
| 47 | \( 1 + 9.24T + 47T^{2} \) |
| 53 | \( 1 + 7.65iT - 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 3.17iT - 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 9.65T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 16.1iT - 83T^{2} \) |
| 89 | \( 1 + 2.34T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−10.37965835808982879777434371122, −9.785304635584572648354327627400, −8.302217867562039739066971583252, −7.81701851002899209713460732231, −6.73614122161069076279188744652, −6.26887304899866316696005646534, −5.05251101358870115694250175479, −3.61556779580791298077273297534, −2.95145589285992764616782277448, −1.45888006183022756313326972294,
1.04407452574058453280408384602, 2.16188283112582154733223292689, 4.23397165496940752623373231852, 4.60549947786170833939981831188, 5.31558943302594466017904167780, 6.77293698087494270644887960098, 7.66021114897178668741216946346, 8.469706518079776698292551020068, 9.474240753983576280240553769871, 9.714004693987941119211912804985
