| L(s) = 1 | − 83.7i·2-s − 4.96e3·4-s − 1.59e4i·7-s + 2.44e5i·8-s − 3.39e5·11-s + 2.02e6i·13-s − 1.33e6·14-s + 1.02e7·16-s − 2.45e6i·17-s + 4.08e6·19-s + 2.84e7i·22-s − 2.86e7i·23-s + 1.69e8·26-s + 7.92e7i·28-s − 9.41e6·29-s + ⋯ |
| L(s) = 1 | − 1.85i·2-s − 2.42·4-s − 0.359i·7-s + 2.63i·8-s − 0.636·11-s + 1.51i·13-s − 0.664·14-s + 2.44·16-s − 0.418i·17-s + 0.378·19-s + 1.17i·22-s − 0.928i·23-s + 2.79·26-s + 0.870i·28-s − 0.0852·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.8714700909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8714700909\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 83.7iT - 2.04e3T^{2} \) |
| 7 | \( 1 + 1.59e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + 3.39e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.02e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + 2.45e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 - 4.08e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 2.86e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + 9.41e6T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.99e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.57e8iT - 1.77e17T^{2} \) |
| 41 | \( 1 + 1.83e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 6.56e8iT - 9.29e17T^{2} \) |
| 47 | \( 1 + 1.97e8iT - 2.47e18T^{2} \) |
| 53 | \( 1 + 5.15e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 6.62e8T + 3.01e19T^{2} \) |
| 61 | \( 1 - 5.58e8T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.01e10iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 1.78e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.33e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 1.24e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.37e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 - 2.94e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.13e11iT - 7.15e21T^{2} \) |
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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
−9.920391278207674358966537276110, −9.060342863020081154943102316899, −8.078827936142013020578228111685, −6.59755964609177548297705814818, −4.92687550801472266179770464588, −4.28995871681384727409124845265, −3.11801442749107035509862255341, −2.23734418909473713907390015537, −1.21571555607082564101283858009, −0.21853291106537972863883562443,
0.887010004766636975769588915160, 2.88557754036166460222395055429, 4.23076729874191125543158815548, 5.45831539857748573303375691473, 5.78781116041827701055388396925, 7.09203923183308873237124391116, 7.893682163345916971886452765043, 8.549167308759117424415933136612, 9.642292675989178980984479611576, 10.57275202777668098411114966030
