| L(s) = 1 | + 8.86i·5-s − 61.8·7-s − 109. i·11-s − 155.·13-s + 395. i·17-s − 140.·19-s + 927. i·23-s + 546.·25-s + 373. i·29-s + 1.04e3·31-s − 548. i·35-s − 194.·37-s + 2.70e3i·41-s − 335.·43-s − 2.85e3i·47-s + ⋯ |
| L(s) = 1 | + 0.354i·5-s − 1.26·7-s − 0.904i·11-s − 0.921·13-s + 1.37i·17-s − 0.388·19-s + 1.75i·23-s + 0.874·25-s + 0.444i·29-s + 1.08·31-s − 0.447i·35-s − 0.142·37-s + 1.61i·41-s − 0.181·43-s − 1.29i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5079160818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5079160818\) |
| \(L(3)\) |
|
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 | \( 1 - 8.86iT - 625T^{2} \) |
| 7 | \( 1 + 61.8T + 2.40e3T^{2} \) |
| 11 | \( 1 + 109. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 155.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 395. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 140.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 927. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 373. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.04e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 194.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.70e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 335.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.85e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.76e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.02e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 7.04e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 6.87e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 821. iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.09e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 7.56e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 7.82e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.28e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 3.78e3T + 8.85e7T^{2} \) |
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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
−8.964715735000073093398642928616, −8.112634244013965948812136441610, −7.17611931653841263829959790283, −6.36372848866730351132345652187, −5.79998512306901883617971053462, −4.60577234061422301718987047048, −3.40438862232397777389208003592, −2.96101001329775537966693847359, −1.53847932034010035131644113338, −0.13975040071162151424431301968,
0.72049502581295292026940942175, 2.34990479489080020224957724414, 2.98329755076207436623175105664, 4.39167846462839249381168357158, 4.89768870181110213291404852390, 6.14278430371522110181582434669, 6.87973351050651030834071717502, 7.50495064923441420943775359368, 8.696795552187081940169093245951, 9.347090954055944915423131646864
