L(s) = 1 | + (−12.3 − 9.51i)3-s + 26.7i·5-s − 70.6i·7-s + (61.8 + 235. i)9-s − 98.7·11-s + 210.·13-s + (254. − 330. i)15-s + 524. i·17-s − 1.11e3i·19-s + (−672. + 872. i)21-s − 866.·23-s + 2.40e3·25-s + (1.47e3 − 3.48e3i)27-s − 2.16e3i·29-s − 1.72e3i·31-s + ⋯ |
L(s) = 1 | + (−0.791 − 0.610i)3-s + 0.478i·5-s − 0.545i·7-s + (0.254 + 0.967i)9-s − 0.246·11-s + 0.345·13-s + (0.292 − 0.379i)15-s + 0.440i·17-s − 0.708i·19-s + (−0.332 + 0.431i)21-s − 0.341·23-s + 0.770·25-s + (0.388 − 0.921i)27-s − 0.477i·29-s − 0.322i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.1094029724\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1094029724\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (12.3 + 9.51i)T \) |
good | 5 | \( 1 - 26.7iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 70.6iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 98.7T + 1.61e5T^{2} \) |
| 13 | \( 1 - 210.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 524. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.11e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 866.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.16e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.72e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 1.23e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.39e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 9.13e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.18e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.60e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.09e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.96e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.75e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.50e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.00e5iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 2.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 619. iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 7.49e4T + 8.58e9T^{2} \) |
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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
−10.68312819134301562241779595455, −10.49007414791724606419105448196, −9.003658741223248931051847749041, −7.83031667272007706898277275971, −7.04721915067751612274867895343, −6.24703722448636979711902638289, −5.20748849343159409705987187907, −4.01374587952467616038516962005, −2.53343514462023167535769986034, −1.19089852990158550651954350856,
0.03357237779973090259945177796, 1.41319706517908120612468827386, 3.13060020705003829704660751804, 4.37597561417074501275011692057, 5.30227235937805374818690033115, 6.05668270203748940254355683340, 7.22133814282449677232422845930, 8.568454337093893246881116181474, 9.237218374944111479518734416318, 10.28951572842015286383929956931