| L(s) = 1 | + (−6.57 − 6.14i)3-s + 9.58i·5-s + 22.6·7-s + (5.37 + 80.8i)9-s + 2.72i·11-s − 220.·13-s + (58.9 − 63.0i)15-s − 172. i·17-s + 275.·19-s + (−149. − 139. i)21-s + 420. i·23-s + 533.·25-s + (461. − 564. i)27-s − 207. i·29-s + 1.08e3·31-s + ⋯ |
| L(s) = 1 | + (−0.730 − 0.683i)3-s + 0.383i·5-s + 0.462·7-s + (0.0663 + 0.997i)9-s + 0.0224i·11-s − 1.30·13-s + (0.262 − 0.280i)15-s − 0.598i·17-s + 0.763·19-s + (−0.337 − 0.316i)21-s + 0.794i·23-s + 0.852·25-s + (0.633 − 0.773i)27-s − 0.246i·29-s + 1.13·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7731812088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7731812088\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (6.57 + 6.14i)T \) |
| good | 5 | \( 1 - 9.58iT - 625T^{2} \) |
| 7 | \( 1 - 22.6T + 2.40e3T^{2} \) |
| 11 | \( 1 - 2.72iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 220.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 172. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 275.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 420. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 207. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.08e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.26e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.23e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.38e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.43e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.63e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.98e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.17e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 5.98e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 7.55e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.32e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 279.T + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.33e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.25e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.65e4T + 8.85e7T^{2} \) |
| 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
−10.45920345975438865877668379728, −9.663282807345577264158606272950, −8.282467557723387831726714211434, −7.33985273978085252925981965244, −6.76262397067484496766704370148, −5.43175310047524225995744227518, −4.75975409726165769234644614339, −2.96164761505810768197139506524, −1.70747800438576031123819714607, −0.26566840469902502268584352157,
1.18039412158737032295344150149, 2.96194349192359792992048308849, 4.48472493947115699245139954992, 4.99787416343137942149609923283, 6.13007898241057322285821683961, 7.22748913908565789503937169337, 8.401046501878429675096012747520, 9.367892154715546571595657867756, 10.22124103892915763627999423359, 10.94595472774876016253199861785
