| L(s) = 1 | + (−11.0 − 11.0i)3-s + (160. + 160. i)5-s + 53.5·7-s + 242. i·9-s + (−122. + 122. i)11-s + (−413. + 413. i)13-s − 3.53e3i·15-s − 3.29e3·17-s + (7.93e3 + 7.93e3i)19-s + (−589. − 589. i)21-s + 1.36e4·23-s + 3.58e4i·25-s + (2.67e3 − 2.67e3i)27-s + (−5.98e3 + 5.98e3i)29-s + 2.31e4i·31-s + ⋯ |
| L(s) = 1 | + (−0.408 − 0.408i)3-s + (1.28 + 1.28i)5-s + 0.156·7-s + 0.333i·9-s + (−0.0919 + 0.0919i)11-s + (−0.188 + 0.188i)13-s − 1.04i·15-s − 0.670·17-s + (1.15 + 1.15i)19-s + (−0.0636 − 0.0636i)21-s + 1.12·23-s + 2.29i·25-s + (0.136 − 0.136i)27-s + (−0.245 + 0.245i)29-s + 0.778i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.014813469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.014813469\) |
| \(L(4)\) |
|
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 | \( 1 + (-160. - 160. i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 - 53.5T + 1.17e5T^{2} \) |
| 11 | \( 1 + (122. - 122. i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (413. - 413. i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + 3.29e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (-7.93e3 - 7.93e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 - 1.36e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (5.98e3 - 5.98e3i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 - 2.31e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (6.29e4 + 6.29e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 - 9.07e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-1.01e5 + 1.01e5i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + 3.26e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (-2.86e4 - 2.86e4i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (1.35e5 - 1.35e5i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (-2.47e5 + 2.47e5i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (5.23e4 + 5.23e4i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 + 3.81e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 1.24e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 2.23e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (5.68e5 + 5.68e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 8.63e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.32e6T + 8.32e11T^{2} \) |
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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
−10.63916461706128688257100435108, −9.883976504019012990521967877308, −8.943031496581770623571358321001, −7.44702998456428416689432389857, −6.85306845724236054948516342935, −5.91381739152436483985773482512, −5.13622103612468369874713980111, −3.39025683128499476768073512544, −2.30384877615312419288209900826, −1.36731987143149014526969957902,
0.46930708843448778792698812611, 1.46375990463377891185311856428, 2.78804060344479638380433684121, 4.51730668211135537984451023005, 5.15595959804884440203245137373, 5.92478249552034731157499809334, 7.10565671752374052836207135308, 8.534576534244726477455678749985, 9.260091684070208515918778603299, 9.844343593665059167222686916586
