| L(s) = 1 | + (−5.65 − 9.79i)2-s + (−63.9 + 110. i)4-s + (−254. + 146. i)5-s + (−1.18e3 + 2.09e3i)7-s + 1.44e3·8-s + (2.87e3 + 1.65e3i)10-s + (9.84e3 − 1.70e4i)11-s − 2.73e4i·13-s + (2.71e4 − 265. i)14-s + (−8.19e3 − 1.41e4i)16-s + (−4.14e4 − 2.39e4i)17-s + (7.22e4 − 4.17e4i)19-s − 3.75e4i·20-s − 2.22e5·22-s + (6.46e4 + 1.11e5i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.406 + 0.234i)5-s + (−0.491 + 0.870i)7-s + 0.353·8-s + (0.287 + 0.165i)10-s + (0.672 − 1.16i)11-s − 0.958i·13-s + (0.707 − 0.00691i)14-s + (−0.125 − 0.216i)16-s + (−0.495 − 0.286i)17-s + (0.554 − 0.320i)19-s − 0.234i·20-s − 0.951·22-s + (0.231 + 0.400i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.8074865188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8074865188\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.65 + 9.79i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.18e3 - 2.09e3i)T \) |
| good | 5 | \( 1 + (254. - 146. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-9.84e3 + 1.70e4i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + 2.73e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (4.14e4 + 2.39e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-7.22e4 + 4.17e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-6.46e4 - 1.11e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 7.10e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (3.84e5 + 2.21e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-1.76e5 - 3.06e5i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 3.55e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 6.52e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-9.65e5 + 5.57e5i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (5.98e6 - 1.03e7i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.01e7 - 5.85e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-1.13e7 + 6.57e6i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (7.88e6 - 1.36e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 2.41e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.51e7 - 8.76e6i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-3.57e6 - 6.18e6i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 6.34e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (7.67e7 - 4.43e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + 6.52e7iT - 7.83e15T^{2} \) |
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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
−11.73064127720768082323449356403, −11.20365113062231078484398489370, −9.886115111541086667489997496225, −8.936495359167889526345694450200, −8.020719271104891398955897539949, −6.58374519411588757997410407179, −5.30792159117313871353108160954, −3.56020307870057045840707030324, −2.75818209649305801200891538947, −1.01174602153322238773382265100,
0.29208571149584557792184630295, 1.72612253862406969939232333168, 3.84892066417393845394132406558, 4.75588755050754188613478301783, 6.53646214396524345681763948434, 7.12298331264489373094760773199, 8.367185228735499917801516994983, 9.493201451262913767206487954792, 10.29676461027065757885142927676, 11.64642942591546925817464937184
