| L(s) = 1 | + (1.23 − 3.80i)2-s + (−7.28 − 5.29i)3-s + (−12.9 − 9.40i)4-s + (55.5 − 6.01i)5-s + (−29.1 + 21.1i)6-s + 98.1·7-s + (−51.7 + 37.6i)8-s + (25.0 + 77.0i)9-s + (45.8 − 218. i)10-s + (−177. + 545. i)11-s + (44.4 + 136. i)12-s + (262. + 807. i)13-s + (121. − 373. i)14-s + (−436. − 250. i)15-s + (79.1 + 243. i)16-s + (1.28e3 − 935. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.994 − 0.107i)5-s + (−0.330 + 0.239i)6-s + 0.757·7-s + (−0.286 + 0.207i)8-s + (0.103 + 0.317i)9-s + (0.144 − 0.692i)10-s + (−0.441 + 1.35i)11-s + (0.0892 + 0.274i)12-s + (0.430 + 1.32i)13-s + (0.165 − 0.509i)14-s + (−0.500 − 0.287i)15-s + (0.0772 + 0.237i)16-s + (1.08 − 0.785i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.147080907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.147080907\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 + 3.80i)T \) |
| 3 | \( 1 + (7.28 + 5.29i)T \) |
| 5 | \( 1 + (-55.5 + 6.01i)T \) |
| good | 7 | \( 1 - 98.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + (177. - 545. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-262. - 807. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-1.28e3 + 935. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (2.34e3 - 1.70e3i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (831. - 2.55e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-4.24e3 - 3.08e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-3.88e3 + 2.82e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (3.03e3 + 9.33e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-1.71e3 - 5.27e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.99e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-8.79e3 - 6.38e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (2.07e4 + 1.50e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.15e4 + 3.56e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-2.99e3 + 9.20e3i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (1.49e4 - 1.08e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-2.46e4 - 1.79e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (6.73e3 - 2.07e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-4.27e4 - 3.10e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (4.62e4 - 3.36e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-5.80e3 + 1.78e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-8.80e4 - 6.39e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−12.20168323101803833899453351152, −11.12841181826457807285267439372, −10.14915386887117627595069225558, −9.317460834022023020883455857519, −7.87437136866720511761923876509, −6.50132331349329727584062601621, −5.32918769879836441496300052002, −4.32924498297286291669558873544, −2.19558674582177859458054449003, −1.41520487564091706845218336958,
0.78311556640221796131337473891, 2.90854705299308667207096787864, 4.63039312375741157419347508776, 5.74440325172786645241472008099, 6.30246183526316798824809377253, 8.065596206326501765556443403651, 8.750912784269838400110106018237, 10.42096556959601562612769738549, 10.73862574890521641499431449281, 12.32643255912511969262236393093
