| L(s) = 1 | − 984. i·5-s − 3.73e3·7-s + 2.64e4i·11-s − 3.41e4·13-s − 5.93e4i·17-s + 1.50e5·19-s + 4.12e5i·23-s − 5.77e5·25-s + 5.98e5i·29-s − 7.83e5·31-s + 3.67e6i·35-s + 2.79e6·37-s − 1.64e6i·41-s + 2.14e6·43-s − 6.13e6i·47-s + ⋯ |
| L(s) = 1 | − 1.57i·5-s − 1.55·7-s + 1.80i·11-s − 1.19·13-s − 0.710i·17-s + 1.15·19-s + 1.47i·23-s − 1.47·25-s + 0.845i·29-s − 0.848·31-s + 2.44i·35-s + 1.49·37-s − 0.583i·41-s + 0.628·43-s − 1.25i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.9143656441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9143656441\) |
| \(L(5)\) |
|
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 | \( 1 + 984. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 3.73e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 2.64e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.41e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 5.93e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.50e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 4.12e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 5.98e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 7.83e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 2.79e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 1.64e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.14e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 6.13e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 9.16e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 5.63e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.03e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.13e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.03e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.81e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 6.33e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 3.67e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 6.35e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 5.70e7T + 7.83e15T^{2} \) |
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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
−9.591885338649046637731990149834, −9.079344231310489796905041293651, −7.50575969286715697775869284188, −7.10619973795924052496606682093, −5.56433342176870467237868520778, −4.93465938918783392030897766285, −3.90668449329020328206838063728, −2.61906087645247876758106356479, −1.40604914247889226192229057407, −0.27649336194331632072275342716,
0.60759241966173453329466014888, 2.65904032594340270079708976528, 2.97781959286850485841188899757, 3.95582352075576826616328101175, 5.76878010813056633705486206226, 6.33000676769435291510407145390, 7.11019009587859536142544124584, 8.105516959553518198422631164809, 9.404375142066865162161268911107, 10.05540489608907230801218140122
