| L(s) = 1 | + 29.2i·2-s − 601.·4-s + 455. i·5-s − 3.24e3·7-s − 1.01e4i·8-s − 1.33e4·10-s − 4.13e3i·11-s + 3.18e4·13-s − 9.51e4i·14-s + 1.42e5·16-s + 1.30e5i·17-s + 2.31e5·19-s − 2.73e5i·20-s + 1.21e5·22-s − 3.11e5i·23-s + ⋯ |
| L(s) = 1 | + 1.83i·2-s − 2.34·4-s + 0.728i·5-s − 1.35·7-s − 2.46i·8-s − 1.33·10-s − 0.282i·11-s + 1.11·13-s − 2.47i·14-s + 2.17·16-s + 1.56i·17-s + 1.77·19-s − 1.71i·20-s + 0.516·22-s − 1.11i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.591800845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.591800845\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 29.2iT - 256T^{2} \) |
| 5 | \( 1 - 455. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 3.24e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 4.13e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 3.18e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.30e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 2.31e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 3.11e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 8.48e4iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 3.42e4T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.61e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 2.85e4iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 3.92e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 7.12e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.34e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 2.34e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 6.76e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.00e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 5.94e5iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.46e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 2.04e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 8.93e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 4.95e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.81e7T + 7.83e15T^{2} \) |
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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
−10.85978831668662616082549325456, −9.861236005008420857124496858995, −8.932198342924320433979530604548, −8.028396005796741765250337473287, −6.94373989533393733954547774579, −6.29230572497735651021715025713, −5.65702709227418301161093113743, −4.06298038371390468716270070207, −3.14097637849220332135197358192, −0.74012937283398982889433331738,
0.60477798758596652714587728236, 1.25057575534621853968560410477, 2.82495057768689049969577911141, 3.45469706679870773974795851136, 4.65106837261781608647994052938, 5.73782194544043827827481644042, 7.41219038863758891981551635340, 8.985035577745596549492993912006, 9.399459800736974025072091419328, 10.13165968241707701893027016731
