| L(s) = 1 | + (−0.0820 − 1.41i)2-s + (−1.09 + 1.33i)3-s + (−1.98 + 0.231i)4-s + (−2.65 + 2.50i)5-s + (1.98 + 1.44i)6-s + (−4.31 + 0.504i)7-s + (0.490 + 2.78i)8-s + (−0.585 − 2.94i)9-s + (3.76 + 3.54i)10-s + (3.61 − 1.08i)11-s + (1.87 − 2.91i)12-s + (−0.0113 + 0.000660i)13-s + (1.06 + 6.05i)14-s + (−0.437 − 6.31i)15-s + (3.89 − 0.920i)16-s + (−1.65 − 0.603i)17-s + ⋯ |
| L(s) = 1 | + (−0.0580 − 0.998i)2-s + (−0.634 + 0.773i)3-s + (−0.993 + 0.115i)4-s + (−1.18 + 1.12i)5-s + (0.808 + 0.588i)6-s + (−1.63 + 0.190i)7-s + (0.173 + 0.984i)8-s + (−0.195 − 0.980i)9-s + (1.18 + 1.12i)10-s + (1.09 − 0.326i)11-s + (0.540 − 0.841i)12-s + (−0.00314 + 0.000183i)13-s + (0.285 + 1.61i)14-s + (−0.112 − 1.63i)15-s + (0.973 − 0.230i)16-s + (−0.401 − 0.146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.179247 - 0.221228i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.179247 - 0.221228i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0820 + 1.41i)T \) |
| 3 | \( 1 + (1.09 - 1.33i)T \) |
| good | 5 | \( 1 + (2.65 - 2.50i)T + (0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (4.31 - 0.504i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (-3.61 + 1.08i)T + (9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (0.0113 - 0.000660i)T + (12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (1.65 + 0.603i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-1.90 - 5.22i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (3.04 + 0.355i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (3.91 + 7.79i)T + (-17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (-2.45 + 3.30i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (1.65 + 1.97i)T + (-6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-3.71 + 2.44i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (0.851 - 3.59i)T + (-38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-0.983 - 1.32i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (-3.28 + 1.89i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.89 + 2.06i)T + (49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (-12.2 - 5.28i)T + (41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (6.50 - 12.9i)T + (-40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (0.647 + 3.67i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.21 + 12.5i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-9.65 - 6.34i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (-3.76 + 5.72i)T + (-32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (-2.58 + 14.6i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (1.51 - 1.60i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.32611142306954332283115853642, −9.758912224216911076252353226168, −9.012508027158143854264348645477, −7.75057182310250935945811147410, −6.52353874922060186960314091217, −5.84649181290942478158676685314, −3.97541913025284149178287056041, −3.83920604460397744452229508643, −2.82690544505370013841369117649, −0.23706983564312433850710799372,
0.932706652611037834316003231205, 3.55841397879977717139073468286, 4.51959218430077016036489005447, 5.49189443504046124951095671191, 6.72671102714442280516403816696, 6.94760755646255965020494493332, 8.026150054705607795974190436994, 8.950242727073229664994214093317, 9.527706947073620632371920442776, 10.85863109631128081730429718220
