| L(s) = 1 | − 2.82i·2-s − 8.00·4-s − 8.92i·5-s + 22.6i·8-s − 25.2·10-s − 112. i·11-s + 86.8·13-s + 64.0·16-s + 321. i·17-s + 103.·19-s + 71.4i·20-s − 318.·22-s − 120. i·23-s + 545.·25-s − 245. i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.357i·5-s + 0.353i·8-s − 0.252·10-s − 0.929i·11-s + 0.514·13-s + 0.250·16-s + 1.11i·17-s + 0.287·19-s + 0.178i·20-s − 0.657·22-s − 0.227i·23-s + 0.872·25-s − 0.363i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.891838565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.891838565\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 8.92iT - 625T^{2} \) |
| 11 | \( 1 + 112. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 86.8T + 2.85e4T^{2} \) |
| 17 | \( 1 - 321. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 103.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 120. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.51e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.36e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 68.3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 96.3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 273.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.22e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.80e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 378. iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.61e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 5.67e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.30e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 9.48e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.73e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.61e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.19e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.05e4T + 8.85e7T^{2} \) |
| 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
−9.412257136745089596115582119838, −8.737400454289889549350945260807, −8.131014615166136466157767250407, −6.88532582788269800281185743137, −5.83016001360458055535323868775, −5.02351334579120067851189097871, −3.84047702431071833086479330336, −3.12773995891806011220000144104, −1.72910183898726194057408536402, −0.801928688453309670222191333549,
0.58671405628671892871058725274, 2.09968404196082110017181090113, 3.36799583991523151025821077557, 4.45323485155609882207993432525, 5.32577777996637680817048792669, 6.30057514480849584864350869620, 7.17042997045790829071654735828, 7.70319699063750045846350277550, 8.821705035124408787595673283532, 9.535903934593452931621507468623
