| L(s) = 1 | + (−7.27 + 8.66i)2-s + (61.3 − 52.8i)3-s + (−22.2 − 126. i)4-s + (209. − 575. i)5-s + (11.6 + 916. i)6-s + (358. − 2.03e3i)7-s + (1.25e3 + 724. i)8-s + (974. − 6.48e3i)9-s + (3.46e3 + 6.00e3i)10-s + (−1.87e3 − 5.15e3i)11-s + (−8.02e3 − 6.56e3i)12-s + (−427. + 358. i)13-s + (1.50e4 + 1.79e4i)14-s + (−1.75e4 − 4.64e4i)15-s + (−1.53e4 + 5.60e3i)16-s + (−6.24e4 + 3.60e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.541i)2-s + (0.757 − 0.652i)3-s + (−0.0868 − 0.492i)4-s + (0.335 − 0.921i)5-s + (0.00902 + 0.707i)6-s + (0.149 − 0.847i)7-s + (0.306 + 0.176i)8-s + (0.148 − 0.988i)9-s + (0.346 + 0.600i)10-s + (−0.128 − 0.352i)11-s + (−0.387 − 0.316i)12-s + (−0.0149 + 0.0125i)13-s + (0.391 + 0.466i)14-s + (−0.347 − 0.917i)15-s + (−0.234 + 0.0855i)16-s + (−0.747 + 0.431i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.796608 - 1.41203i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.796608 - 1.41203i\) |
| \(L(5)\) |
|
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 | \( 1 + (-61.3 + 52.8i)T \) |
| good | 5 | \( 1 + (-209. + 575. i)T + (-2.99e5 - 2.51e5i)T^{2} \) |
| 7 | \( 1 + (-358. + 2.03e3i)T + (-5.41e6 - 1.97e6i)T^{2} \) |
| 11 | \( 1 + (1.87e3 + 5.15e3i)T + (-1.64e8 + 1.37e8i)T^{2} \) |
| 13 | \( 1 + (427. - 358. i)T + (1.41e8 - 8.03e8i)T^{2} \) |
| 17 | \( 1 + (6.24e4 - 3.60e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (7.58e4 - 1.31e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.21e4 - 2.13e3i)T + (7.35e10 - 2.67e10i)T^{2} \) |
| 29 | \( 1 + (-8.39e4 + 1.00e5i)T + (-8.68e10 - 4.92e11i)T^{2} \) |
| 31 | \( 1 + (1.73e5 + 9.81e5i)T + (-8.01e11 + 2.91e11i)T^{2} \) |
| 37 | \( 1 + (1.08e6 + 1.88e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (1.67e6 + 1.99e6i)T + (-1.38e12 + 7.86e12i)T^{2} \) |
| 43 | \( 1 + (-6.15e6 + 2.23e6i)T + (8.95e12 - 7.51e12i)T^{2} \) |
| 47 | \( 1 + (4.51e6 + 7.96e5i)T + (2.23e13 + 8.14e12i)T^{2} \) |
| 53 | \( 1 - 8.48e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (6.50e6 - 1.78e7i)T + (-1.12e14 - 9.43e13i)T^{2} \) |
| 61 | \( 1 + (-2.29e6 + 1.29e7i)T + (-1.80e14 - 6.55e13i)T^{2} \) |
| 67 | \( 1 + (6.02e5 - 5.05e5i)T + (7.05e13 - 3.99e14i)T^{2} \) |
| 71 | \( 1 + (-1.88e7 + 1.09e7i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-7.76e6 + 1.34e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (9.26e5 + 7.77e5i)T + (2.63e14 + 1.49e15i)T^{2} \) |
| 83 | \( 1 + (2.24e7 - 2.67e7i)T + (-3.91e14 - 2.21e15i)T^{2} \) |
| 89 | \( 1 + (3.19e7 + 1.84e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-1.53e8 + 5.58e7i)T + (6.00e15 - 5.03e15i)T^{2} \) |
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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
−13.43360418108254106600159446576, −12.46251353535880286585531490271, −10.65940713249854165799543412949, −9.271218017616400379226158801473, −8.379421539393993463059557129765, −7.32494548805297188244645495569, −5.92814411728430691072249707169, −4.10023068024876704941242425394, −1.84783164661767118269197651940, −0.58120833063872636679651112862,
2.15536812923151655185871410371, 3.02345419198701472339144020113, 4.79291797278654960541144421432, 6.83624734959954511398562380681, 8.399244252159150337040212157323, 9.357862502130049599415539347044, 10.42922892675678476681619906373, 11.36137324203627852004312137072, 12.90508905753477050879664736246, 14.12637559947098129624843350431
