| L(s) = 1 | + (−496. − 860. i)5-s + (−302. − 174. i)7-s + (2.00e4 + 1.15e4i)11-s + (−1.34e4 − 2.33e4i)13-s − 1.24e5·17-s − 3.17e4i·19-s + (3.30e5 − 1.90e5i)23-s + (−2.98e5 + 5.17e5i)25-s + (−7.41e3 + 1.28e4i)29-s + (−4.19e4 + 2.42e4i)31-s + 3.46e5i·35-s − 1.19e6·37-s + (3.42e5 + 5.93e5i)41-s + (1.17e6 + 6.76e5i)43-s + (−6.77e6 − 3.91e6i)47-s + ⋯ |
| L(s) = 1 | + (−0.795 − 1.37i)5-s + (−0.125 − 0.0726i)7-s + (1.36 + 0.789i)11-s + (−0.472 − 0.818i)13-s − 1.48·17-s − 0.243i·19-s + (1.18 − 0.681i)23-s + (−0.764 + 1.32i)25-s + (−0.0104 + 0.0181i)29-s + (−0.0454 + 0.0262i)31-s + 0.231i·35-s − 0.639·37-s + (0.121 + 0.210i)41-s + (0.342 + 0.197i)43-s + (−1.38 − 0.801i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0938 - 0.995i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.0938 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.2689236343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2689236343\) |
| \(L(5)\) |
|
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 | \( 1 + (496. + 860. i)T + (-1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (302. + 174. i)T + (2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-2.00e4 - 1.15e4i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + (1.34e4 + 2.33e4i)T + (-4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 + 1.24e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 3.17e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (-3.30e5 + 1.90e5i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (7.41e3 - 1.28e4i)T + (-2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (4.19e4 - 2.42e4i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + 1.19e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-3.42e5 - 5.93e5i)T + (-3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-1.17e6 - 6.76e5i)T + (5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (6.77e6 + 3.91e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + 1.50e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + (8.40e6 - 4.84e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-7.01e6 + 1.21e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.44e7 + 8.31e6i)T + (2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 1.37e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.77e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-2.12e7 - 1.22e7i)T + (7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-8.78e5 - 5.07e5i)T + (1.12e15 + 1.95e15i)T^{2} \) |
| 89 | \( 1 + 1.06e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (-4.23e7 + 7.34e7i)T + (-3.91e15 - 6.78e15i)T^{2} \) |
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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
−9.811508301135395041587303024230, −8.990369383423622044155976573671, −8.418900195827940341284290642285, −7.29158411516772379668950868259, −6.45429564507644475942525236925, −4.89653435912868225515485713627, −4.54927195834547277028695168389, −3.40168632994324001741668011388, −1.87143122579404770089145692255, −0.799024480501204470519380809023,
0.06388262242831796610000153696, 1.59380659838365404668244196346, 2.88923028711440974087849686462, 3.66701195829902009183267513951, 4.61218549962673821416483574173, 6.27946069340943478669190100401, 6.74564893989566143089299082128, 7.58101537218203903108169027986, 8.808039354787776851748415112480, 9.492945072779069676805761025460
