L(s) = 1 | + 173. i·5-s + 484·7-s − 1.34e3i·11-s − 3.36e3·13-s − 12.7i·17-s + 5.74e3·19-s − 3.37e3i·23-s − 1.46e4·25-s + 2.93e4i·29-s + 3.97e4·31-s + 8.41e4i·35-s − 5.25e4·37-s + 3.70e4i·41-s + 3.80e3·43-s + 7.67e4i·47-s + ⋯ |
L(s) = 1 | + 1.39i·5-s + 1.41·7-s − 1.00i·11-s − 1.53·13-s − 0.00259i·17-s + 0.837·19-s − 0.277i·23-s − 0.936·25-s + 1.20i·29-s + 1.33·31-s + 1.96i·35-s − 1.03·37-s + 0.537i·41-s + 0.0477·43-s + 0.739i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.888005982\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.888005982\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 173. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 484T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.34e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.36e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 12.7iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.74e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 3.37e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.93e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.97e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 5.25e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 3.70e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 3.80e3T + 6.32e9T^{2} \) |
| 47 | \( 1 - 7.67e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.38e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.49e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.32e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 1.68e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 5.31e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.36e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 3.51e4T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.09e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.29e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.21e5T + 8.32e11T^{2} \) |
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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
−10.35022171726521897036781096684, −9.213317446180268905437375063074, −8.098031324176693798511009557961, −7.44934367639735212854745379499, −6.61276169954335565554319859192, −5.45123449337657301546702552530, −4.59737755004628403176798868479, −3.19666628802339647221024496809, −2.46850392260194968053101770445, −1.16467125975890339456892349164,
0.38529768320131408649471333716, 1.48884625047421493810732870659, 2.31743698091971696467434712122, 4.14766820006059447331224570061, 5.00064836253169559777266452057, 5.24583512857765705917418361888, 6.97745736850846790714074030286, 7.86546637511547730109050287042, 8.435574006494326632353156161892, 9.543951896866464836713259394027