| L(s) = 1 | + (−0.493 + 0.414i)2-s + (−1.31 + 7.46i)4-s + (−0.483 + 0.175i)5-s + (−1.66 − 9.45i)7-s + (−5.02 − 8.69i)8-s + (0.165 − 0.287i)10-s + (−38.1 − 13.9i)11-s + (−1.34 − 1.12i)13-s + (4.74 + 3.97i)14-s + (−50.9 − 18.5i)16-s + (37.5 − 65.1i)17-s + (42.1 + 73.0i)19-s + (−0.677 − 3.84i)20-s + (24.6 − 8.95i)22-s + (22.4 − 127. i)23-s + ⋯ |
| L(s) = 1 | + (−0.174 + 0.146i)2-s + (−0.164 + 0.933i)4-s + (−0.0432 + 0.0157i)5-s + (−0.0900 − 0.510i)7-s + (−0.221 − 0.384i)8-s + (0.00524 − 0.00907i)10-s + (−1.04 − 0.381i)11-s + (−0.0286 − 0.0240i)13-s + (0.0905 + 0.0759i)14-s + (−0.795 − 0.289i)16-s + (0.536 − 0.929i)17-s + (0.508 + 0.881i)19-s + (−0.00757 − 0.0429i)20-s + (0.238 − 0.0868i)22-s + (0.203 − 1.15i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.602440 - 0.521270i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.602440 - 0.521270i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (0.493 - 0.414i)T + (1.38 - 7.87i)T^{2} \) |
| 5 | \( 1 + (0.483 - 0.175i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (1.66 + 9.45i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (38.1 + 13.9i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (1.34 + 1.12i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-37.5 + 65.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-42.1 - 73.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-22.4 + 127. i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-200. + 168. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-52.5 + 297. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (103. - 178. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (240. + 201. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (122. + 44.5i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (34.5 + 195. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + 527.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (72.6 - 26.4i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (46.5 + 263. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-68.9 - 57.8i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-360. + 624. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-66.7 - 115. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-97.0 + 81.4i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (647. - 543. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-693. - 1.20e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-265. - 96.7i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−11.61724481525750617680081220178, −10.36545822798339296899031113202, −9.538266747674233644649327354702, −8.177696641936866939718764017352, −7.74735975400920030797486777357, −6.56702919895598622218303391639, −5.13213006892656890848254342957, −3.83153622695456791834066884432, −2.69416420859044093207113343360, −0.34594828112622575699367954817,
1.47416837156719542356699297103, 2.95976412853377857978901548843, 4.82965594028912524217755231486, 5.58831065492745373001768182895, 6.78163811996626548969025385991, 8.106513953898480693070070136873, 9.089113905336449900970513980937, 10.07157319333147351691332306308, 10.70086706636996750478962773428, 11.82468266340105303085807125029
