| L(s) = 1 | + (7.27 − 8.66i)2-s + (−76.9 + 25.1i)3-s + (−22.2 − 126. i)4-s + (421. − 1.15e3i)5-s + (−341. + 850. i)6-s + (−165. + 937. i)7-s + (−1.25e3 − 724. i)8-s + (5.29e3 − 3.87e3i)9-s + (−6.96e3 − 1.20e4i)10-s + (−1.20e3 − 3.31e3i)11-s + (4.88e3 + 9.14e3i)12-s + (2.44e4 − 2.05e4i)13-s + (6.91e3 + 8.24e3i)14-s + (−3.27e3 + 9.96e4i)15-s + (−1.53e4 + 5.60e3i)16-s + (−8.20e4 + 4.73e4i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (−0.950 + 0.310i)3-s + (−0.0868 − 0.492i)4-s + (0.673 − 1.85i)5-s + (−0.263 + 0.656i)6-s + (−0.0688 + 0.390i)7-s + (−0.306 − 0.176i)8-s + (0.806 − 0.591i)9-s + (−0.696 − 1.20i)10-s + (−0.0825 − 0.226i)11-s + (0.235 + 0.440i)12-s + (0.855 − 0.717i)13-s + (0.180 + 0.214i)14-s + (−0.0647 + 1.96i)15-s + (−0.234 + 0.0855i)16-s + (−0.982 + 0.567i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0972007 + 1.23735i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0972007 + 1.23735i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.27 + 8.66i)T \) |
| 3 | \( 1 + (76.9 - 25.1i)T \) |
| good | 5 | \( 1 + (-421. + 1.15e3i)T + (-2.99e5 - 2.51e5i)T^{2} \) |
| 7 | \( 1 + (165. - 937. i)T + (-5.41e6 - 1.97e6i)T^{2} \) |
| 11 | \( 1 + (1.20e3 + 3.31e3i)T + (-1.64e8 + 1.37e8i)T^{2} \) |
| 13 | \( 1 + (-2.44e4 + 2.05e4i)T + (1.41e8 - 8.03e8i)T^{2} \) |
| 17 | \( 1 + (8.20e4 - 4.73e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-4.97e4 + 8.62e4i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (4.73e5 - 8.34e4i)T + (7.35e10 - 2.67e10i)T^{2} \) |
| 29 | \( 1 + (2.47e5 - 2.95e5i)T + (-8.68e10 - 4.92e11i)T^{2} \) |
| 31 | \( 1 + (-1.36e5 - 7.72e5i)T + (-8.01e11 + 2.91e11i)T^{2} \) |
| 37 | \( 1 + (9.49e5 + 1.64e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-9.60e5 - 1.14e6i)T + (-1.38e12 + 7.86e12i)T^{2} \) |
| 43 | \( 1 + (-1.96e6 + 7.14e5i)T + (8.95e12 - 7.51e12i)T^{2} \) |
| 47 | \( 1 + (1.52e6 + 2.68e5i)T + (2.23e13 + 8.14e12i)T^{2} \) |
| 53 | \( 1 + 5.10e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (6.95e6 - 1.91e7i)T + (-1.12e14 - 9.43e13i)T^{2} \) |
| 61 | \( 1 + (-4.65e6 + 2.63e7i)T + (-1.80e14 - 6.55e13i)T^{2} \) |
| 67 | \( 1 + (-1.26e7 + 1.06e7i)T + (7.05e13 - 3.99e14i)T^{2} \) |
| 71 | \( 1 + (1.46e7 - 8.44e6i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (1.24e7 - 2.15e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-2.32e7 - 1.95e7i)T + (2.63e14 + 1.49e15i)T^{2} \) |
| 83 | \( 1 + (-4.57e7 + 5.45e7i)T + (-3.91e14 - 2.21e15i)T^{2} \) |
| 89 | \( 1 + (3.94e7 + 2.27e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (9.43e7 - 3.43e7i)T + (6.00e15 - 5.03e15i)T^{2} \) |
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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
−12.82289105966801995819595407283, −12.11455752121183591114139318506, −10.84120560534521377858541789621, −9.599014518042901858663153022191, −8.585829949516769315157033067192, −6.02621512984439236728738590781, −5.30527153557864507156758707914, −4.12673528423166762276872238221, −1.64960207423197928283419709087, −0.41269008338249187991376937497,
2.14443562550067043895376429121, 3.99864516019777268139554457741, 5.92078186184846482541973959957, 6.58317225023893518819144821950, 7.55090944932694686180012082600, 9.867759024436916672481215309665, 10.90139482001569140030664232022, 11.81598291597697802360038867237, 13.52012119870093442117439393203, 13.99154257163845500745944460511
