| L(s) = 1 | + 275. i·3-s − 1.39e3·5-s + 1.93e4i·7-s − 1.69e4·9-s − 1.71e5i·11-s − 2.03e5·13-s − 3.85e5i·15-s + 2.12e6·17-s − 5.59e5i·19-s − 5.32e6·21-s − 8.97e6i·23-s + 1.95e6·25-s + 1.16e7i·27-s + 6.60e6·29-s + 4.19e7i·31-s + ⋯ |
| L(s) = 1 | + 1.13i·3-s − 0.447·5-s + 1.14i·7-s − 0.286·9-s − 1.06i·11-s − 0.547·13-s − 0.507i·15-s + 1.49·17-s − 0.225i·19-s − 1.30·21-s − 1.39i·23-s + 0.200·25-s + 0.809i·27-s + 0.321·29-s + 1.46i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.444438827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.444438827\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 1.39e3T \) |
| good | 3 | \( 1 - 275. iT - 5.90e4T^{2} \) |
| 7 | \( 1 - 1.93e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 1.71e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 2.03e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 2.12e6T + 2.01e12T^{2} \) |
| 19 | \( 1 + 5.59e5iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 8.97e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 6.60e6T + 4.20e14T^{2} \) |
| 31 | \( 1 - 4.19e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + 3.19e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 1.21e8T + 1.34e16T^{2} \) |
| 43 | \( 1 - 1.35e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 3.08e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 7.27e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 2.51e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 9.40e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 2.07e9iT - 1.82e18T^{2} \) |
| 71 | \( 1 - 1.90e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.72e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 2.36e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 5.22e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 8.10e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 5.70e9T + 7.37e19T^{2} \) |
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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
−10.37422807174465973466500758727, −9.421486742450033702680752904535, −8.690296269074348185186963755556, −7.80667454371743911327862543291, −6.37437872875038215133155066933, −5.35754866462319426530228429798, −4.61962403069831600188220834601, −3.40509233062208732250150936900, −2.72313050641462847904929305442, −1.05196427281786520305828426670,
0.30296470800731249388223365622, 1.17345266875237697435003007284, 2.05789738707977438651804381665, 3.48638649949375591604271120825, 4.43919492743569473829263969101, 5.70219733842295511059462025241, 7.04697612348856119720756793333, 7.41422568616067345473752937601, 8.030295841022454663321528209330, 9.654542105005653192940066567537
