| L(s) = 1 | + 23.7i·5-s + 9.38i·7-s + 112.·11-s − 56.5i·13-s + 79.9·17-s − 211.·19-s + 217. i·23-s + 60.9·25-s + 616. i·29-s + 1.11e3i·31-s − 222.·35-s − 802. i·37-s + 2.41e3·41-s + 2.13e3·43-s − 3.59e3i·47-s + ⋯ |
| L(s) = 1 | + 0.949i·5-s + 0.191i·7-s + 0.925·11-s − 0.334i·13-s + 0.276·17-s − 0.585·19-s + 0.410i·23-s + 0.0975·25-s + 0.733i·29-s + 1.15i·31-s − 0.181·35-s − 0.586i·37-s + 1.43·41-s + 1.15·43-s − 1.62i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.192002672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192002672\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 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 | \( 1 + 3.11e3T + 8.85e7T^{2} \) |
| 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
−9.334215167709222311328243109175, −8.784087304554264533001359560223, −7.65309367227334555131596469418, −6.94520217342022664191736808015, −6.20965038141777980913909375669, −5.30324151189394985783375635522, −4.08275598746746876691920221647, −3.24896400034309482322589595626, −2.26407887754839867262785340611, −1.00783873622038003180693953130,
0.53440382642205387233339586166, 1.41442815346879677606749980562, 2.64095562430086824599247618689, 4.11262653880234729208411879645, 4.46663609320199515640909983888, 5.72491855253912988402846497198, 6.42199036269978686603071609765, 7.48721954787110435272777582547, 8.284737175588774136303409679439, 9.150884447659773249252004688534
