| L(s) = 1 | − 5.26i·2-s + 4.31·4-s − 131. i·7-s − 191. i·8-s − 290.·11-s + 68.3i·13-s − 689.·14-s − 867.·16-s − 310. i·17-s + 2.13e3·19-s + 1.52e3i·22-s − 873. i·23-s + 359.·26-s − 564. i·28-s − 2.58e3·29-s + ⋯ |
| L(s) = 1 | − 0.930i·2-s + 0.134·4-s − 1.01i·7-s − 1.05i·8-s − 0.722·11-s + 0.112i·13-s − 0.940·14-s − 0.847·16-s − 0.260i·17-s + 1.35·19-s + 0.672i·22-s − 0.344i·23-s + 0.104·26-s − 0.136i·28-s − 0.569·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.335920286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.335920286\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 5.26iT - 32T^{2} \) |
| 7 | \( 1 + 131. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 290.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 68.3iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 310. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.13e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 873. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.99e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.69e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.80e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.48e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 7.65e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 9.23e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.32e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.23e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 3.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.65e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.96e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.18e4iT - 8.58e9T^{2} \) |
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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
−10.82212541825111991175558440796, −10.15642369539610528885831368134, −9.193774551426279837060268553267, −7.61298065891296350495504165162, −6.98500188717081961901163137984, −5.44615516391365637469751392367, −4.01552092017270692675135101419, −3.01137345919572233803921218485, −1.63031512402403091216764594750, −0.36253687392018965373720377955,
1.90195304862801877968033341289, 3.18189289056459947679117846628, 5.18786885288904411912424500898, 5.68837685233168168709218745577, 6.93664464903077159493270715862, 7.84790651644835155374383816609, 8.722329036404880551242681896712, 9.826015377994463397562005397684, 11.11708731302893309919701940913, 11.85179931988837039766052030133
