L(s) = 1 | + (−1.5 − 2.59i)2-s + (−0.5 + 0.866i)4-s + (9 + 15.5i)5-s − 21·8-s + (27 − 46.7i)10-s + (−18 + 31.1i)11-s + 34·13-s + (35.5 + 61.4i)16-s + (−21 + 36.3i)17-s + (−62 − 107. i)19-s − 18.0·20-s + 108·22-s + (−99.5 + 172. i)25-s + (−51 − 88.3i)26-s − 102·29-s + ⋯ |
L(s) = 1 | + (−0.530 − 0.918i)2-s + (−0.0625 + 0.108i)4-s + (0.804 + 1.39i)5-s − 0.928·8-s + (0.853 − 1.47i)10-s + (−0.493 + 0.854i)11-s + 0.725·13-s + (0.554 + 0.960i)16-s + (−0.299 + 0.518i)17-s + (−0.748 − 1.29i)19-s − 0.201·20-s + 1.04·22-s + (−0.796 + 1.37i)25-s + (−0.384 − 0.666i)26-s − 0.653·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5612293397\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5612293397\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.5 + 2.59i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-9 - 15.5i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (18 - 31.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 34T + 2.19e3T^{2} \) |
| 17 | \( 1 + (21 - 36.3i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (62 + 107. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 102T + 2.43e4T^{2} \) |
| 31 | \( 1 + (80 - 138. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (199 + 344. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 318T + 6.89e4T^{2} \) |
| 43 | \( 1 + 268T + 7.95e4T^{2} \) |
| 47 | \( 1 + (120 + 207. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (249 - 431. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-66 + 114. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-199 - 344. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (46 - 79.6i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 720T + 3.57e5T^{2} \) |
| 73 | \( 1 + (251 - 434. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-512 - 886. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 204T + 5.71e5T^{2} \) |
| 89 | \( 1 + (177 + 306. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 286T + 9.12e5T^{2} \) |
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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
−10.73783619237473958207888739322, −10.35750882434308464184856416558, −9.456545376130456516394900418410, −8.593546229540331771165171535595, −7.06954791184809672158301334054, −6.46526332292883172555140169716, −5.35915765392515151868403210278, −3.61128950994451395853673891524, −2.50271471630768336240804782313, −1.77620100138586813884053069954,
0.19888443130704633358617067793, 1.72437727461189320406063306224, 3.45759249559648530847212653071, 5.02618390217363575956534735582, 5.83515999586349296442664760332, 6.58114522080549294565524718101, 8.079313455730280518434742586027, 8.411322810560292227992972926212, 9.245959524958729161291051045576, 10.08460477547032023170846694478