| L(s) = 1 | + (−14.5 + 6.70i)2-s + 150. i·3-s + (166. − 194. i)4-s + (−1.00e3 − 2.18e3i)6-s − 2.62e3i·7-s + (−1.10e3 + 3.94e3i)8-s − 1.60e4·9-s + 2.30e3i·11-s + (2.92e4 + 2.49e4i)12-s − 4.70e4·13-s + (1.76e4 + 3.81e4i)14-s + (−1.03e4 − 6.47e4i)16-s + 5.19e4·17-s + (2.32e5 − 1.07e5i)18-s − 5.95e4i·19-s + ⋯ |
| L(s) = 1 | + (−0.907 + 0.419i)2-s + 1.85i·3-s + (0.648 − 0.760i)4-s + (−0.777 − 1.68i)6-s − 1.09i·7-s + (−0.270 + 0.962i)8-s − 2.43·9-s + 0.157i·11-s + (1.41 + 1.20i)12-s − 1.64·13-s + (0.458 + 0.993i)14-s + (−0.158 − 0.987i)16-s + 0.622·17-s + (2.21 − 1.02i)18-s − 0.456i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.792216 + 0.365635i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.792216 + 0.365635i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (14.5 - 6.70i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 150. iT - 6.56e3T^{2} \) |
| 7 | \( 1 + 2.62e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 2.30e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.70e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 5.19e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 5.95e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 7.75e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 9.02e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 3.40e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 5.84e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.93e5T + 7.98e12T^{2} \) |
| 43 | \( 1 - 2.95e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 5.03e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 7.54e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 8.82e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.08e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.44e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 3.71e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.62e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 4.88e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 6.93e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 1.05e8T + 3.93e15T^{2} \) |
| 97 | \( 1 + 1.33e8T + 7.83e15T^{2} \) |
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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
−11.91563291879040726688798542018, −10.74596861057210218845205394557, −10.09204402165492544838993508195, −9.498408683945851957493327317032, −8.242296532564898208780104813299, −6.97481188278234146241091790434, −5.31687561136950507823940712973, −4.38615596953797159112476514807, −2.78884474387455338189662803213, −0.47326522905783900801981031410,
0.78766195038337765790283962735, 2.08030983016381558323837946605, 2.80797000784135027411534825911, 5.66305192594944747228616368539, 6.85419519817625512332209814561, 7.76885284058738307222438477980, 8.588259318371215458196201898023, 9.805790724362889351006671395888, 11.42725776017963674391722883001, 12.30574290906991795550623469219
