| L(s) = 1 | − 3.25i·3-s − 2.23·5-s − 2.64i·7-s − 7.62·9-s + 0.359i·11-s + 5.64·13-s + 7.28i·15-s − 7.99·17-s − 8.62·21-s + 5.00·25-s + 15.0i·27-s − 0.623·29-s + 1.17·33-s + 5.91i·35-s − 18.4i·39-s + ⋯ |
| L(s) = 1 | − 1.88i·3-s − 0.999·5-s − 0.999i·7-s − 2.54·9-s + 0.108i·11-s + 1.56·13-s + 1.88i·15-s − 1.93·17-s − 1.88·21-s + 1.00·25-s + 2.90i·27-s − 0.115·29-s + 0.204·33-s + 0.999i·35-s − 2.94i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.191344 + 0.714107i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.191344 + 0.714107i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 7 | \( 1 + 2.64iT \) |
| good | 3 | \( 1 + 3.25iT - 3T^{2} \) |
| 11 | \( 1 - 0.359iT - 11T^{2} \) |
| 13 | \( 1 - 5.64T + 13T^{2} \) |
| 17 | \( 1 + 7.99T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 0.623T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 5.71iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 8.00iT - 79T^{2} \) |
| 83 | \( 1 + 15.8iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
| 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
−10.74488619062780293174136988515, −8.849980988498817313281393842431, −8.330353613028930614805361462156, −7.41227033782638691596019662694, −6.83174371204105711978369794949, −6.08417031724781065281165918963, −4.42283328646971430529917934844, −3.25917129190739408353162167940, −1.71866435994139693634324893874, −0.42311400585236984447762164787,
2.78001999655357903604867306239, 3.83351106868102674910036791202, 4.49018042332909453534590621514, 5.53574716862512571044365703774, 6.51220815346085600193467990414, 8.277237647675474773538033311005, 8.744828710519556153881368192666, 9.358897218087170466855333661502, 10.58305402184725682327201629128, 11.22298839174559025149280020389
