| 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
−14.12637559947098129624843350431, −12.90508905753477050879664736246, −11.36137324203627852004312137072, −10.42922892675678476681619906373, −9.357862502130049599415539347044, −8.399244252159150337040212157323, −6.83624734959954511398562380681, −4.79291797278654960541144421432, −3.02345419198701472339144020113, −2.15536812923151655185871410371,
0.58120833063872636679651112862, 1.84783164661767118269197651940, 4.10023068024876704941242425394, 5.92814411728430691072249707169, 7.32494548805297188244645495569, 8.379421539393993463059557129765, 9.271218017616400379226158801473, 10.65940713249854165799543412949, 12.46251353535880286585531490271, 13.43360418108254106600159446576
