| L(s) = 1 | + 0.464i·5-s + 10.8·7-s − 15.9i·11-s − 17.9·13-s + 26.1i·17-s − 3.55·19-s − 4.12i·23-s + 24.7·25-s − 41.6i·29-s + 7.04·31-s + 5.01i·35-s − 9.85·37-s − 43.5i·41-s − 35.5·43-s − 16.1i·47-s + ⋯ |
| L(s) = 1 | + 0.0928i·5-s + 1.54·7-s − 1.44i·11-s − 1.38·13-s + 1.53i·17-s − 0.187·19-s − 0.179i·23-s + 0.991·25-s − 1.43i·29-s + 0.227·31-s + 0.143i·35-s − 0.266·37-s − 1.06i·41-s − 0.827·43-s − 0.343i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.871164564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.871164564\) |
| \(L(2)\) |
|
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 | \( 1 - 0.464iT - 25T^{2} \) |
| 7 | \( 1 - 10.8T + 49T^{2} \) |
| 11 | \( 1 + 15.9iT - 121T^{2} \) |
| 13 | \( 1 + 17.9T + 169T^{2} \) |
| 17 | \( 1 - 26.1iT - 289T^{2} \) |
| 19 | \( 1 + 3.55T + 361T^{2} \) |
| 23 | \( 1 + 4.12iT - 529T^{2} \) |
| 29 | \( 1 + 41.6iT - 841T^{2} \) |
| 31 | \( 1 - 7.04T + 961T^{2} \) |
| 37 | \( 1 + 9.85T + 1.36e3T^{2} \) |
| 41 | \( 1 + 43.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 35.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 16.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 75.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 28.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 59.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 23.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 37.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 52.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 120.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 116. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 135. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94.4T + 9.40e3T^{2} \) |
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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
−8.399154664433958835018778499835, −7.83106758942532764518823886563, −6.94496406498710545430097589805, −6.00542547904899800271233886076, −5.28885776544158847110804221863, −4.54366964945896450737261423308, −3.67631417293706436737380326646, −2.52414559151673289849799690412, −1.66168051401636510041218375408, −0.43069172751414102183851265965,
1.17097561675383888707657879367, 2.10651720462965557197836530747, 2.92931571191026632812265726189, 4.47293436122722252041172966412, 4.85082659086569444019860122320, 5.31071437285020612688285968721, 6.86151494773992941541161876988, 7.24398209052427071449669150340, 7.907920398205823741566800391194, 8.762141407232993870389820556752
