| L(s) = 1 | + (−0.299 + 1.97i)2-s + (−3.82 − 1.18i)4-s + (−3.75 + 1.36i)5-s + (12.2 − 2.15i)7-s + (3.48 − 7.20i)8-s + (−1.58 − 7.83i)10-s + (−5.05 + 13.8i)11-s + (8.77 + 7.35i)13-s + (0.606 + 24.7i)14-s + (13.2 + 9.03i)16-s + (−2.73 + 4.73i)17-s + (−20.0 + 11.5i)19-s + (15.9 − 0.782i)20-s + (−25.9 − 14.1i)22-s + (−15.3 − 2.70i)23-s + ⋯ |
| L(s) = 1 | + (−0.149 + 0.988i)2-s + (−0.955 − 0.295i)4-s + (−0.751 + 0.273i)5-s + (1.74 − 0.307i)7-s + (0.435 − 0.900i)8-s + (−0.158 − 0.783i)10-s + (−0.459 + 1.26i)11-s + (0.674 + 0.566i)13-s + (0.0433 + 1.77i)14-s + (0.825 + 0.564i)16-s + (−0.160 + 0.278i)17-s + (−1.05 + 0.608i)19-s + (0.798 − 0.0391i)20-s + (−1.17 − 0.642i)22-s + (−0.666 − 0.117i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.489i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.288335 + 1.10207i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.288335 + 1.10207i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.299 - 1.97i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.75 - 1.36i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-12.2 + 2.15i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (5.05 - 13.8i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-8.77 - 7.35i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (2.73 - 4.73i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (20.0 - 11.5i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (15.3 + 2.70i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-16.7 + 14.0i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (11.7 + 2.07i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (6.45 - 11.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-55.7 - 46.7i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (20.3 - 55.8i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (20.2 - 3.56i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 16.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-6.75 - 18.5i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-5.25 - 29.8i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (52.6 - 62.8i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-36.4 - 21.0i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (44.1 + 76.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-38.2 - 45.5i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-1.27 - 1.52i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (45.8 + 79.3i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-64.3 - 23.4i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.67460640901437631629861335869, −10.82561841915235568007827234613, −9.874493631565300352783314687974, −8.491569853518233587768679391754, −7.948122614924068236184439022759, −7.22665345675906789186678094780, −6.02032545347155035087889237995, −4.59309646946578698903749197413, −4.19382766760322553851702628910, −1.67542336339192311669453229994,
0.60349002752586995079946336265, 2.17021495599639461531596160216, 3.67159638982382749441442162733, 4.71331445083208004568290529205, 5.66991797100043351403087653665, 7.70401124854569817976382799296, 8.425697157756959670168371079107, 8.820769861627605528021155892194, 10.57242035872191520567574937270, 11.02597467466354811707324945269
