| L(s) = 1 | + (3.92 + 0.772i)2-s + (14.8 + 6.06i)4-s − 21.1·5-s + 44.6i·7-s + (53.4 + 35.2i)8-s + (−83.0 − 16.3i)10-s + 67.7i·11-s − 29.1·13-s + (−34.4 + 175. i)14-s + (182. + 179. i)16-s − 402.·17-s − 644. i·19-s + (−313. − 128. i)20-s + (−52.3 + 265. i)22-s + 387. i·23-s + ⋯ |
| L(s) = 1 | + (0.981 + 0.193i)2-s + (0.925 + 0.378i)4-s − 0.846·5-s + 0.910i·7-s + (0.834 + 0.550i)8-s + (−0.830 − 0.163i)10-s + 0.560i·11-s − 0.172·13-s + (−0.175 + 0.893i)14-s + (0.712 + 0.701i)16-s − 1.39·17-s − 1.78i·19-s + (−0.782 − 0.320i)20-s + (−0.108 + 0.549i)22-s + 0.732i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.697475418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.697475418\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.92 - 0.772i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 21.1T + 625T^{2} \) |
| 7 | \( 1 - 44.6iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 67.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 29.1T + 2.85e4T^{2} \) |
| 17 | \( 1 + 402.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 644. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 387. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 724.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.25e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.40e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.54e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.87e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 4.16e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 906.T + 7.89e6T^{2} \) |
| 59 | \( 1 + 4.52e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.62e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 67.7iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 1.31e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 9.47e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.36e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 762. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 8.08e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 6.66e3T + 8.85e7T^{2} \) |
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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
−11.46803744998676280667671090252, −10.99334310535335921651909628488, −9.394631744156219344354283596874, −8.413916988605105607348121153319, −7.30230606781921794967154225318, −6.56589823427426057198775973711, −5.21904242567715281104924092563, −4.46229334156339687118396321136, −3.19016853668829180034192365297, −2.04408342353128453326031808357,
0.33090443199650256754111572355, 2.01511125521932001144552502659, 3.69154065495586022562391089080, 4.09779421966819445986193763155, 5.46105014390535428453102148384, 6.62258927793904769240005398838, 7.48899615870080101718156586738, 8.451537313849835163541730785541, 10.04189311990037685826651272731, 10.79244651287169930403500708305
