| 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
| \(L(1)\) |
\(\approx\) |
\(0.487426 - 1.17675i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.487426 - 1.17675i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.65T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.657448048839029545109967041801, −9.123464818259849779094121022842, −8.188790718295408240349613117596, −7.58313225320253415606373861551, −6.11826974937664386379332394320, −5.65152041045524100176233676347, −4.36752970342008527072918730975, −3.64647125349208478407920517947, −1.81276725017448476451303767819, −0.62829978778964559294913902009,
2.03461265030526628840788154398, 3.11613623200854586013292907791, 4.03187814717639336184021727139, 5.19086641391245261929584303840, 6.54197576658128428865201221991, 7.05221444555958398399666801898, 7.57128541178665485449744828465, 9.139560203743679339736399502575, 9.907703271192412412982055206239, 10.41109927303224836865226226631
