| L(s) = 1 | + (−1 + 2.82i)3-s + 5.65i·5-s + 6·7-s + (−7.00 − 5.65i)9-s − 5.65i·11-s + 10·13-s + (−16.0 − 5.65i)15-s − 22.6i·17-s − 2·19-s + (−6 + 16.9i)21-s + 11.3i·23-s − 7.00·25-s + (23.0 − 14.1i)27-s + 16.9i·29-s + 22·31-s + ⋯ |
| L(s) = 1 | + (−0.333 + 0.942i)3-s + 1.13i·5-s + 0.857·7-s + (−0.777 − 0.628i)9-s − 0.514i·11-s + 0.769·13-s + (−1.06 − 0.377i)15-s − 1.33i·17-s − 0.105·19-s + (−0.285 + 0.808i)21-s + 0.491i·23-s − 0.280·25-s + (0.851 − 0.523i)27-s + 0.585i·29-s + 0.709·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.867701 + 0.613557i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.867701 + 0.613557i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 2.82i)T \) |
| good | 5 | \( 1 - 5.65iT - 25T^{2} \) |
| 7 | \( 1 - 6T + 49T^{2} \) |
| 11 | \( 1 + 5.65iT - 121T^{2} \) |
| 13 | \( 1 - 10T + 169T^{2} \) |
| 17 | \( 1 + 22.6iT - 289T^{2} \) |
| 19 | \( 1 + 2T + 361T^{2} \) |
| 23 | \( 1 - 11.3iT - 529T^{2} \) |
| 29 | \( 1 - 16.9iT - 841T^{2} \) |
| 31 | \( 1 - 22T + 961T^{2} \) |
| 37 | \( 1 + 6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 82T + 1.84e3T^{2} \) |
| 47 | \( 1 + 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 62.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 73.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 86T + 3.72e3T^{2} \) |
| 67 | \( 1 + 2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 82T + 5.32e3T^{2} \) |
| 79 | \( 1 + 10T + 6.24e3T^{2} \) |
| 83 | \( 1 - 73.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 33.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94T + 9.40e3T^{2} \) |
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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
−15.57043223853227462553880449452, −14.63570113284640684849674349504, −13.74779223154144646161891469990, −11.59147277940091212263005169755, −11.06628828728236644478383214993, −9.903901932416096870875403707062, −8.409602321260687544304099349977, −6.66243153714466825535656970955, −5.10001899190853530745711742341, −3.29890576054193911342403457106,
1.51516477393485154652426285439, 4.69649591590057934138421818576, 6.15665602481661924731916389292, 7.898390187659691057364673114554, 8.733397303057901341913019031213, 10.69952286346587240119431300380, 11.97297937000266713434486219270, 12.79875184868878539502629179602, 13.80983076217042636039033038115, 15.18205616010965058745776826752
