| L(s) = 1 | + (1.22 − 1.58i)2-s + (−1.02 − 3.86i)4-s + (−4.29 + 1.56i)5-s + (11.3 − 1.99i)7-s + (−7.37 − 3.10i)8-s + (−2.76 + 8.70i)10-s + (2.79 − 7.68i)11-s + (−18.9 − 15.8i)13-s + (10.6 − 20.4i)14-s + (−13.9 + 7.89i)16-s + (6.56 − 11.3i)17-s + (8.74 − 5.04i)19-s + (10.4 + 15.0i)20-s + (−8.76 − 13.8i)22-s + (−20.3 − 3.58i)23-s + ⋯ |
| L(s) = 1 | + (0.610 − 0.792i)2-s + (−0.255 − 0.966i)4-s + (−0.858 + 0.312i)5-s + (1.61 − 0.285i)7-s + (−0.921 − 0.388i)8-s + (−0.276 + 0.870i)10-s + (0.254 − 0.698i)11-s + (−1.45 − 1.22i)13-s + (0.762 − 1.45i)14-s + (−0.869 + 0.493i)16-s + (0.386 − 0.668i)17-s + (0.460 − 0.265i)19-s + (0.520 + 0.750i)20-s + (−0.398 − 0.627i)22-s + (−0.884 − 0.155i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.552213 - 1.70958i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.552213 - 1.70958i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 + 1.58i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (4.29 - 1.56i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-11.3 + 1.99i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-2.79 + 7.68i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (18.9 + 15.8i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-6.56 + 11.3i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-8.74 + 5.04i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (20.3 + 3.58i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (6.33 - 5.31i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (9.50 + 1.67i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-21.4 + 37.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-33.7 - 28.3i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (3.42 - 9.42i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-87.0 + 15.3i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 8.52T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-4.23 - 11.6i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (5.96 + 33.8i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-38.0 + 45.3i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-22.0 - 12.7i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (9.11 + 15.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-48.7 - 58.0i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (20.4 + 24.3i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-13.1 - 22.7i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (29.5 + 10.7i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.20706255802604549644469818093, −10.48467099582032349810573255263, −9.360890705571305808177796297978, −7.984765871727619860928136534897, −7.40999431398398006169732693916, −5.63393289847017635381298582746, −4.82242495310246449304352902225, −3.74479011193497131761027090371, −2.46639245433673185175622786303, −0.71030787842852691175350415953,
2.09354413514097445616150047771, 4.12827681662958421043093592764, 4.60646597630250952617918798641, 5.69524659215646416516333531076, 7.21917496038196949921698237077, 7.75577719550221071084316859254, 8.582528716088148143208623652598, 9.687452506665460903312713178151, 11.28821945578725288046809452810, 12.16098435409773918581141367262
