| L(s) = 1 | + (−11.0 − 11.0i)3-s + (15.5 + 15.5i)5-s − 10.2·7-s + 242. i·9-s + (1.30e3 − 1.30e3i)11-s + (−1.99e3 + 1.99e3i)13-s − 343. i·15-s − 1.80e3·17-s + (1.70e3 + 1.70e3i)19-s + (113. + 113. i)21-s − 5.64e3·23-s − 1.51e4i·25-s + (2.67e3 − 2.67e3i)27-s + (1.17e4 − 1.17e4i)29-s + 3.22e4i·31-s + ⋯ |
| L(s) = 1 | + (−0.408 − 0.408i)3-s + (0.124 + 0.124i)5-s − 0.0300·7-s + 0.333i·9-s + (0.984 − 0.984i)11-s + (−0.910 + 0.910i)13-s − 0.101i·15-s − 0.366·17-s + (0.248 + 0.248i)19-s + (0.0122 + 0.0122i)21-s − 0.464·23-s − 0.968i·25-s + (0.136 − 0.136i)27-s + (0.482 − 0.482i)29-s + 1.08i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1769371843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1769371843\) |
| \(L(4)\) |
|
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 | \( 1 + (-15.5 - 15.5i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 + 10.2T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-1.30e3 + 1.30e3i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (1.99e3 - 1.99e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + 1.80e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (-1.70e3 - 1.70e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 + 5.64e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-1.17e4 + 1.17e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 - 3.22e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (-4.95e4 - 4.95e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 + 8.73e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-7.12e3 + 7.12e3i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + 7.94e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (5.80e4 + 5.80e4i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (-9.26e3 + 9.26e3i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (3.12e4 - 3.12e4i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (-1.40e5 - 1.40e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 + 4.45e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 6.74e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 1.71e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-7.09e5 - 7.09e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 5.49e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 6.75e5T + 8.32e11T^{2} \) |
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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
−9.867826991387225817692106524684, −8.924883439247839320553889018730, −7.967058691124542488603877281782, −6.74912963517680878152969664883, −6.26318735953696932411120728664, −5.00336422222625568992521355530, −3.89284489409910111576664849892, −2.49605762633823774046733185184, −1.29228226351157226299865349955, −0.04363107796435315440697414526,
1.33625664179979411428294242577, 2.73431240228494284732696238378, 4.10933865183148044127635031578, 4.94460885213051579532597822101, 5.99751504386823216563328934063, 7.03698735129677632466280147072, 7.964648696587639501252337395377, 9.413704212954640808718743399489, 9.675147237859950260650708593867, 10.82578211528822163642487246650
