| L(s) = 1 | + (−1.81 + 2.16i)2-s + (−0.346 + 8.99i)3-s + (−1.38 − 7.87i)4-s + (−2.28 + 6.26i)5-s + (−18.8 − 17.1i)6-s + (−10.8 + 61.4i)7-s + (19.5 + 11.3i)8-s + (−80.7 − 6.23i)9-s + (−9.42 − 16.3i)10-s + (−78.0 − 214. i)11-s + (71.3 − 9.76i)12-s + (16.0 − 13.4i)13-s + (−113. − 135. i)14-s + (−55.5 − 22.6i)15-s + (−60.1 + 21.8i)16-s + (−356. + 205. i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.541i)2-s + (−0.0385 + 0.999i)3-s + (−0.0868 − 0.492i)4-s + (−0.0912 + 0.250i)5-s + (−0.523 − 0.475i)6-s + (−0.220 + 1.25i)7-s + (0.306 + 0.176i)8-s + (−0.997 − 0.0769i)9-s + (−0.0942 − 0.163i)10-s + (−0.644 − 1.77i)11-s + (0.495 − 0.0677i)12-s + (0.0951 − 0.0798i)13-s + (−0.578 − 0.689i)14-s + (−0.246 − 0.100i)15-s + (−0.234 + 0.0855i)16-s + (−1.23 + 0.711i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.110077 - 0.622860i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.110077 - 0.622860i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.81 - 2.16i)T \) |
| 3 | \( 1 + (0.346 - 8.99i)T \) |
| good | 5 | \( 1 + (2.28 - 6.26i)T + (-478. - 401. i)T^{2} \) |
| 7 | \( 1 + (10.8 - 61.4i)T + (-2.25e3 - 821. i)T^{2} \) |
| 11 | \( 1 + (78.0 + 214. i)T + (-1.12e4 + 9.41e3i)T^{2} \) |
| 13 | \( 1 + (-16.0 + 13.4i)T + (4.95e3 - 2.81e4i)T^{2} \) |
| 17 | \( 1 + (356. - 205. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (93.7 - 162. i)T + (-6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-436. + 77.0i)T + (2.62e5 - 9.57e4i)T^{2} \) |
| 29 | \( 1 + (764. - 911. i)T + (-1.22e5 - 6.96e5i)T^{2} \) |
| 31 | \( 1 + (-136. - 773. i)T + (-8.67e5 + 3.15e5i)T^{2} \) |
| 37 | \( 1 + (-969. - 1.67e3i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (441. + 525. i)T + (-4.90e5 + 2.78e6i)T^{2} \) |
| 43 | \( 1 + (1.30e3 - 473. i)T + (2.61e6 - 2.19e6i)T^{2} \) |
| 47 | \( 1 + (-580. - 102. i)T + (4.58e6 + 1.66e6i)T^{2} \) |
| 53 | \( 1 - 2.25e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-858. + 2.35e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-674. + 3.82e3i)T + (-1.30e7 - 4.73e6i)T^{2} \) |
| 67 | \( 1 + (559. - 469. i)T + (3.49e6 - 1.98e7i)T^{2} \) |
| 71 | \( 1 + (3.48e3 - 2.01e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (4.96e3 - 8.60e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (832. + 698. i)T + (6.76e6 + 3.83e7i)T^{2} \) |
| 83 | \( 1 + (-5.75e3 + 6.86e3i)T + (-8.24e6 - 4.67e7i)T^{2} \) |
| 89 | \( 1 + (-6.76e3 - 3.90e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-1.16e4 + 4.25e3i)T + (6.78e7 - 5.69e7i)T^{2} \) |
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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
−15.40498098617581671998688001666, −14.58858086581639725866132604456, −13.11967792537787906876882018901, −11.32508553625346998000635976021, −10.56975491216019546586688646351, −8.973368355267208482333481298084, −8.477586996552822180448226889751, −6.27756141635466600514676174403, −5.24249033845235029958440549478, −3.10010797272515785984187446312,
0.41661059562039890521333075049, 2.26172958089463848908430719902, 4.50880586546687200195853076411, 6.91285884785651102020973876337, 7.63377541612029490629449575103, 9.209848634727470135029209548153, 10.53206056630283061066179283299, 11.67145118484743116703510864204, 12.99133966009476418155896587105, 13.37708603431820664303728981902
