| L(s) = 1 | − 24i·2-s + 1.47e3·4-s + 1.67e4i·7-s − 8.44e4i·8-s − 5.34e5·11-s − 5.77e5i·13-s + 4.01e5·14-s + 9.87e5·16-s − 6.90e6i·17-s − 1.06e7·19-s + 1.28e7i·22-s − 1.86e7i·23-s − 1.38e7·26-s + 2.46e7i·28-s + 1.28e8·29-s + ⋯ |
| L(s) = 1 | − 0.530i·2-s + 0.718·4-s + 0.376i·7-s − 0.911i·8-s − 1.00·11-s − 0.431i·13-s + 0.199·14-s + 0.235·16-s − 1.17i·17-s − 0.987·19-s + 0.530i·22-s − 0.603i·23-s − 0.228·26-s + 0.270i·28-s + 1.16·29-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
| \(L(6)\) |
\(\approx\) |
\(0.06981128106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06981128106\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 24iT - 2.04e3T^{2} \) |
| 7 | \( 1 - 1.67e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + 5.34e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 5.77e5iT - 1.79e12T^{2} \) |
| 17 | \( 1 + 6.90e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 + 1.06e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.86e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 - 1.28e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 5.28e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 1.82e8iT - 1.77e17T^{2} \) |
| 41 | \( 1 + 3.08e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.71e7iT - 9.29e17T^{2} \) |
| 47 | \( 1 - 2.68e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 - 1.59e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 5.18e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 6.95e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.54e10iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 9.79e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.46e9iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 3.81e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.93e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 + 2.49e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 7.50e10iT - 7.15e21T^{2} \) |
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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
−10.61792105709686980182465100156, −9.936283860459419913211570258474, −8.674689439542522832357770919741, −7.63529369793335355681269616128, −6.65639521650024516712248753959, −5.60406201079782991377381774993, −4.42261216643305898851799266767, −2.90450685462546244542919448052, −2.47825163432391381323086690244, −1.12808692560075699559936570275,
0.01188972895001192993101294083, 1.58088423171914209355537115453, 2.50092343032972517655854274289, 3.80397859402501414126561527450, 5.09817895199712814539951895883, 6.11389705859075046504692260448, 6.96480550976476341252107537567, 7.912691193284548743896185801747, 8.683487524329814754665786790131, 10.24616687341048772272428663671
