| L(s) = 1 | − 491. i·5-s − 907.·7-s + 5.99e3i·11-s − 4.75e4·13-s + 1.06e5i·17-s + 1.37e5·19-s − 8.00e4i·23-s + 1.49e5·25-s − 4.63e4i·29-s + 9.41e5·31-s + 4.45e5i·35-s − 1.33e6·37-s + 4.74e6i·41-s + 2.35e6·43-s − 2.73e6i·47-s + ⋯ |
| L(s) = 1 | − 0.786i·5-s − 0.377·7-s + 0.409i·11-s − 1.66·13-s + 1.27i·17-s + 1.05·19-s − 0.285i·23-s + 0.381·25-s − 0.0655i·29-s + 1.01·31-s + 0.297i·35-s − 0.714·37-s + 1.68i·41-s + 0.689·43-s − 0.560i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.553847754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.553847754\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 907.T \) |
| good | 5 | \( 1 + 491. iT - 3.90e5T^{2} \) |
| 11 | \( 1 - 5.99e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.75e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.06e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.37e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 8.00e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 4.63e4iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 9.41e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.33e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 4.74e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.35e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 2.73e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.15e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.65e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.01e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.21e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 5.40e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.94e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 7.34e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 3.14e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 8.79e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.00e7T + 7.83e15T^{2} \) |
| 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
−10.19327030511222061673477772293, −9.634371921453102052861785861326, −8.541278139898016907404720436369, −7.58096949863360543968847190859, −6.51228146448594236439003451485, −5.23355036963949235485319360520, −4.45600417210411086864690987633, −3.04146152495154436739060702790, −1.75207335415656024824734129879, −0.46981552953427539776727856197,
0.74489617864481587611052330940, 2.50624304680029110306206195468, 3.16687664246013313157396373698, 4.68789552884567789893164990869, 5.73774554686384898124693486935, 7.06703209980506708353627268619, 7.47101001139487454784886052507, 9.024770482511249919192690017422, 9.833484377122306621757661011625, 10.67474903546536278275245651699
