| L(s) = 1 | + (11.0 + 11.0i)3-s + (−145. − 145. i)5-s − 535.·7-s + 242. i·9-s + (−1.23e3 + 1.23e3i)11-s + (−1.76e3 + 1.76e3i)13-s − 3.20e3i·15-s − 5.53e3·17-s + (−3.22e3 − 3.22e3i)19-s + (−5.90e3 − 5.90e3i)21-s + 2.83e3·23-s + 2.66e4i·25-s + (−2.67e3 + 2.67e3i)27-s + (−1.17e4 + 1.17e4i)29-s + 3.11e4i·31-s + ⋯ |
| L(s) = 1 | + (0.408 + 0.408i)3-s + (−1.16 − 1.16i)5-s − 1.56·7-s + 0.333i·9-s + (−0.928 + 0.928i)11-s + (−0.803 + 0.803i)13-s − 0.950i·15-s − 1.12·17-s + (−0.470 − 0.470i)19-s + (−0.637 − 0.637i)21-s + 0.232·23-s + 1.70i·25-s + (−0.136 + 0.136i)27-s + (−0.480 + 0.480i)29-s + 1.04i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.2184554072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2184554072\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-11.0 - 11.0i)T \) |
| good | 5 | \( 1 + (145. + 145. i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 + 535.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (1.23e3 - 1.23e3i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (1.76e3 - 1.76e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + 5.53e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (3.22e3 + 3.22e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 - 2.83e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (1.17e4 - 1.17e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 - 3.11e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (3.00e4 + 3.00e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 + 2.42e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-3.09e4 + 3.09e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + 6.26e3iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (1.98e5 + 1.98e5i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (-1.34e5 + 1.34e5i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (1.03e5 - 1.03e5i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (3.96e5 + 3.96e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 3.34e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 4.36e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 3.50e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (4.60e5 + 4.60e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 + 1.37e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.47e5T + 8.32e11T^{2} \) |
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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
−10.10780672711993347103609304566, −9.218131794423751237262236336943, −8.710832761341932981291348038739, −7.48002227809045509387214186103, −6.76696318920656591139115759678, −5.06390271307055657381713888828, −4.41072818564778835776394260531, −3.39594982694023812665937111208, −2.13862972793381214387864369191, −0.16864026413398566952113815091,
0.25905559134598247847039640181, 2.66706652996644815021408306462, 3.04874723981663982888451476493, 4.09986374608215753976027994195, 5.89932167955886752558315706021, 6.72694972700472839469641562712, 7.56089369435711447698641180230, 8.272793224976944034201958043339, 9.518796576027619374619032517655, 10.47559984647912755411842469320
