| L(s) = 1 | + (1.51 + 1.30i)2-s + (0.570 + 3.95i)4-s + (−2.52 + 0.918i)5-s + (−1.85 + 0.326i)7-s + (−4.32 + 6.73i)8-s + (−5.01 − 1.91i)10-s + (−4.30 + 11.8i)11-s + (−7.78 − 6.53i)13-s + (−3.22 − 1.93i)14-s + (−15.3 + 4.51i)16-s + (−11.3 + 19.7i)17-s + (18.9 − 10.9i)19-s + (−5.07 − 9.46i)20-s + (−22.0 + 12.2i)22-s + (−1.40 − 0.247i)23-s + ⋯ |
| L(s) = 1 | + (0.755 + 0.654i)2-s + (0.142 + 0.989i)4-s + (−0.504 + 0.183i)5-s + (−0.264 + 0.0466i)7-s + (−0.540 + 0.841i)8-s + (−0.501 − 0.191i)10-s + (−0.391 + 1.07i)11-s + (−0.598 − 0.502i)13-s + (−0.230 − 0.137i)14-s + (−0.959 + 0.282i)16-s + (−0.669 + 1.15i)17-s + (0.996 − 0.575i)19-s + (−0.253 − 0.473i)20-s + (−1.00 + 0.557i)22-s + (−0.0610 − 0.0107i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.185696 + 1.49395i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.185696 + 1.49395i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.51 - 1.30i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (2.52 - 0.918i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (1.85 - 0.326i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (4.30 - 11.8i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (7.78 + 6.53i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (11.3 - 19.7i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-18.9 + 10.9i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (1.40 + 0.247i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (20.6 - 17.3i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-32.6 - 5.75i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-18.4 + 31.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (1.18 + 0.997i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (7.41 - 20.3i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-70.2 + 12.3i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 45.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-31.4 - 86.3i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-13.6 - 77.3i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-69.7 + 83.1i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (29.5 + 17.0i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-20.6 - 35.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (10.0 + 11.9i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-49.5 - 59.0i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (88.0 + 152. i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (39.0 + 14.2i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−12.07305857412925624286839334036, −11.11300701070325598457675382309, −9.966329384797979638742843205489, −8.797532063880510155638200034159, −7.62932615933733091433679838671, −7.14226541950710627711601808220, −5.88996480198990504368307481231, −4.83007017235739744111357820710, −3.78942682413498012217611609611, −2.49273618426857930626519691022,
0.53282432188422364894275227876, 2.48362456629328548023617781979, 3.63824416189254848542740123855, 4.75107222506714221835476750009, 5.78521912390632675787231128575, 6.91827075847258757325551205774, 8.123279628822753048593842016450, 9.395702743424104452204718030483, 10.11071439962410062890412971179, 11.45259818573124475387411956332
