| L(s) = 1 | + (7.27 − 8.66i)2-s + (30.8 + 74.8i)3-s + (−22.2 − 126. i)4-s + (187. − 513. i)5-s + (873. + 276. i)6-s + (−775. + 4.39e3i)7-s + (−1.25e3 − 724. i)8-s + (−4.65e3 + 4.62e3i)9-s + (−3.09e3 − 5.35e3i)10-s + (−1.40e3 − 3.85e3i)11-s + (8.75e3 − 5.55e3i)12-s + (−1.48e4 + 1.24e4i)13-s + (3.24e4 + 3.87e4i)14-s + (4.42e4 − 1.86e3i)15-s + (−1.53e4 + 5.60e3i)16-s + (8.97e4 − 5.18e4i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (0.381 + 0.924i)3-s + (−0.0868 − 0.492i)4-s + (0.299 − 0.822i)5-s + (0.674 + 0.213i)6-s + (−0.322 + 1.83i)7-s + (−0.306 − 0.176i)8-s + (−0.709 + 0.705i)9-s + (−0.309 − 0.535i)10-s + (−0.0958 − 0.263i)11-s + (0.422 − 0.268i)12-s + (−0.518 + 0.435i)13-s + (0.845 + 1.00i)14-s + (0.874 − 0.0369i)15-s + (−0.234 + 0.0855i)16-s + (1.07 − 0.620i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0683 - 0.997i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0683 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.35719 + 1.45341i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35719 + 1.45341i\) |
| \(L(5)\) |
|
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 | \( 1 + (-30.8 - 74.8i)T \) |
| good | 5 | \( 1 + (-187. + 513. i)T + (-2.99e5 - 2.51e5i)T^{2} \) |
| 7 | \( 1 + (775. - 4.39e3i)T + (-5.41e6 - 1.97e6i)T^{2} \) |
| 11 | \( 1 + (1.40e3 + 3.85e3i)T + (-1.64e8 + 1.37e8i)T^{2} \) |
| 13 | \( 1 + (1.48e4 - 1.24e4i)T + (1.41e8 - 8.03e8i)T^{2} \) |
| 17 | \( 1 + (-8.97e4 + 5.18e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.17e5 - 2.03e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.39e5 - 2.45e4i)T + (7.35e10 - 2.67e10i)T^{2} \) |
| 29 | \( 1 + (3.66e5 - 4.36e5i)T + (-8.68e10 - 4.92e11i)T^{2} \) |
| 31 | \( 1 + (-2.71e5 - 1.54e6i)T + (-8.01e11 + 2.91e11i)T^{2} \) |
| 37 | \( 1 + (-1.12e6 - 1.95e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (3.31e6 + 3.95e6i)T + (-1.38e12 + 7.86e12i)T^{2} \) |
| 43 | \( 1 + (-2.41e6 + 8.80e5i)T + (8.95e12 - 7.51e12i)T^{2} \) |
| 47 | \( 1 + (-6.74e6 - 1.18e6i)T + (2.23e13 + 8.14e12i)T^{2} \) |
| 53 | \( 1 + 9.71e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (2.73e6 - 7.50e6i)T + (-1.12e14 - 9.43e13i)T^{2} \) |
| 61 | \( 1 + (3.74e5 - 2.12e6i)T + (-1.80e14 - 6.55e13i)T^{2} \) |
| 67 | \( 1 + (-8.75e6 + 7.34e6i)T + (7.05e13 - 3.99e14i)T^{2} \) |
| 71 | \( 1 + (-5.23e6 + 3.02e6i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-1.90e7 + 3.29e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (1.72e7 + 1.44e7i)T + (2.63e14 + 1.49e15i)T^{2} \) |
| 83 | \( 1 + (-3.02e6 + 3.60e6i)T + (-3.91e14 - 2.21e15i)T^{2} \) |
| 89 | \( 1 + (7.31e6 + 4.22e6i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-5.35e7 + 1.94e7i)T + (6.00e15 - 5.03e15i)T^{2} \) |
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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
−14.05010576435618555246843128694, −12.51436964954659235730093465531, −11.94550450571719927897468677815, −10.27843474795272566869730806371, −9.267011426318175592035523236885, −8.491223124125208679198069553123, −5.76732637173604404000212615095, −5.01457832461848556349516364668, −3.31759894177991294763997888793, −1.98721456761584310916131246306,
0.54344446707692782547095444268, 2.61810637353338520584852118534, 4.05992248348343776200902257012, 6.18702294908066301225746867598, 7.14572358709317984087919009652, 7.87513248773061581535021754535, 9.826444610027161771602654195983, 11.05030948892874545213719882721, 12.71202210721749701205718725392, 13.46755186028223101631771306735
