| L(s) = 1 | + (−1.74 + 0.976i)2-s + (2.09 − 3.40i)4-s + (7.02 − 2.55i)5-s + (8.30 − 1.46i)7-s + (−0.325 + 7.99i)8-s + (−9.76 + 11.3i)10-s + (3.00 − 8.26i)11-s + (5.39 + 4.53i)13-s + (−13.0 + 10.6i)14-s + (−7.23 − 14.2i)16-s + (10.7 − 18.6i)17-s + (−22.4 + 12.9i)19-s + (5.99 − 29.3i)20-s + (2.81 + 17.3i)22-s + (−40.0 − 7.05i)23-s + ⋯ |
| L(s) = 1 | + (−0.872 + 0.488i)2-s + (0.523 − 0.852i)4-s + (1.40 − 0.511i)5-s + (1.18 − 0.209i)7-s + (−0.0406 + 0.999i)8-s + (−0.976 + 1.13i)10-s + (0.273 − 0.751i)11-s + (0.415 + 0.348i)13-s + (−0.933 + 0.762i)14-s + (−0.452 − 0.891i)16-s + (0.634 − 1.09i)17-s + (−1.18 + 0.682i)19-s + (0.299 − 1.46i)20-s + (0.128 + 0.788i)22-s + (−1.74 − 0.306i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.58140 - 0.206082i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58140 - 0.206082i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.74 - 0.976i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-7.02 + 2.55i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-8.30 + 1.46i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-3.00 + 8.26i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-5.39 - 4.53i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-10.7 + 18.6i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (22.4 - 12.9i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (40.0 + 7.05i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-6.59 + 5.53i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-11.2 - 1.98i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-6.02 + 10.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-57.3 - 48.1i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (5.67 - 15.5i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (10.2 - 1.80i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 6.32T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-16.9 - 46.4i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-0.604 - 3.42i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-40.3 + 48.1i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-8.73 - 5.04i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (13.0 + 22.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (61.6 + 73.4i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-2.03 - 2.41i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-23.2 - 40.2i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (8.79 + 3.20i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.11512343023931322243318998642, −10.20803035103526397514247842057, −9.437650714866097271986554755905, −8.508591559674743184919171258491, −7.83323413321812339139674229738, −6.32259039712242751777254077959, −5.74829956824138974761377933698, −4.57969596060168903944007677208, −2.18889105747701550443287627370, −1.14714783182792106844632848172,
1.62356311543827749923382726547, 2.33714984696644211181879018993, 4.10361144081131383132492678590, 5.71365299183858915389310607066, 6.64336042314965719086842242257, 7.900035804009518553985528903194, 8.674970410336763331630467423642, 9.781641641849441116274022960526, 10.39385382105917646238885374831, 11.13255228689459671812587074512
