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
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.06053763825\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06053763825\) |
\(L(3)\) |
|
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 | \( 1 + 1.65e4T + 8.85e7T^{2} \) |
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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
−10.98641199297128391586312959588, −10.30147555398716407191278721659, −9.472611316343708447118687356500, −8.711790170289734939239104148097, −7.57086648176846723235095972753, −6.74552294075573726925379538973, −5.29185905315381200002062009579, −4.31261422319690688761361821377, −3.10636078383973240934170243329, −2.22936180719508515386425263104,
0.01426244226729586722237205582, 1.55796590773195879480148859158, 2.73115910812982909319393413833, 3.90891046067411554837731863077, 5.25705126225992689065521420448, 6.53231957454063805701800218971, 7.32936196611337413828639699910, 8.278569008010120344363840919570, 9.161982297897429299138603448424, 9.889260787092964058130541793540