L(s) = 1 | + (−1.02 − 1.71i)2-s + (−1.88 + 3.52i)4-s + (−2.52 + 0.918i)5-s + (1.85 − 0.326i)7-s + (7.99 − 0.377i)8-s + (4.16 + 3.38i)10-s + (4.30 − 11.8i)11-s + (−7.78 − 6.53i)13-s + (−2.46 − 2.84i)14-s + (−8.85 − 13.3i)16-s + (−11.3 + 19.7i)17-s + (−18.9 + 10.9i)19-s + (1.53 − 10.6i)20-s + (−24.7 + 4.76i)22-s + (1.40 + 0.247i)23-s + ⋯ |
L(s) = 1 | + (−0.513 − 0.858i)2-s + (−0.472 + 0.881i)4-s + (−0.504 + 0.183i)5-s + (0.264 − 0.0466i)7-s + (0.998 − 0.0472i)8-s + (0.416 + 0.338i)10-s + (0.391 − 1.07i)11-s + (−0.598 − 0.502i)13-s + (−0.175 − 0.203i)14-s + (−0.553 − 0.832i)16-s + (−0.669 + 1.15i)17-s + (−0.996 + 0.575i)19-s + (0.0765 − 0.531i)20-s + (−1.12 + 0.216i)22-s + (0.0610 + 0.0107i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00682104 + 0.0144420i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00682104 + 0.0144420i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.02 + 1.71i)T \) |
| 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
−11.35555342099358132665045834517, −11.01674843796731286461712408557, −10.02805941963914157856749191992, −8.867623305991884524835976218660, −8.203078557177495522881470410387, −7.24537753212492879104499833134, −5.79687687441720940270176456822, −4.23057554845486839074237727791, −3.37106772846652201293555992833, −1.82241658551832543379983340763,
0.008665928766000698504131503584, 2.01515964451900427468547995351, 4.31302590485061829867899242691, 4.94949536589942671971624239133, 6.47800390784433552939306804588, 7.22559868474437420084679927443, 8.103539670241523307852398248173, 9.192396222934070226311251660981, 9.756451880897470539503460152467, 11.02732511573953536048142189146