| L(s) = 1 | + 5.19·3-s + 23.7i·5-s + 9.38i·7-s + 27·9-s + 112.·11-s + 56.5i·13-s + 123. i·15-s − 79.9·17-s + 211.·19-s + 48.7i·21-s − 217. i·23-s + 60.9·25-s + 140.·27-s + 616. i·29-s + 1.11e3i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.949i·5-s + 0.191i·7-s + 0.333·9-s + 0.925·11-s + 0.334i·13-s + 0.548i·15-s − 0.276·17-s + 0.585·19-s + 0.110i·21-s − 0.410i·23-s + 0.0975·25-s + 0.192·27-s + 0.733i·29-s + 1.15i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.424844402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.424844402\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.19T \) |
| good | 5 | \( 1 - 23.7iT - 625T^{2} \) |
| 7 | \( 1 - 9.38iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 112.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 56.5iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 79.9T + 8.35e4T^{2} \) |
| 19 | \( 1 - 211.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 217. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 616. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.11e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 802. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 2.41e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.13e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.59e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 833. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 1.30e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 4.78e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 4.02e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 9.48e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 266.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.75e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 7.28e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.41e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 3.11e3T + 8.85e7T^{2} \) |
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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
−10.90754814602470694943227624291, −9.993513852370850104789693381151, −9.095188898399938889216828239284, −8.253318083335678438664421992213, −6.97987518647944833558894981259, −6.54448444717919703345153141689, −5.00851313288749300022041694762, −3.69302260942821602468342415591, −2.80475059434199333864809438717, −1.45610442659753952856946419237,
0.66433513914027309151302908042, 1.88129499231358913351950896578, 3.44811617809558209244076337450, 4.43186729425289537676782113832, 5.53026992363951367245454793024, 6.79108720600460566884285770764, 7.83498252283340755928651440371, 8.728414757847540727440486895663, 9.394837143786732249701081079058, 10.29431315859739386719565609124
