| L(s) = 1 | + (−1.28 − 1.08i)2-s + (0.143 + 0.812i)4-s + (−1.06 − 0.386i)5-s + (−0.678 + 3.84i)7-s + (−0.987 + 1.70i)8-s + (0.949 + 1.64i)10-s + (−1.75 + 0.639i)11-s + (−0.561 + 0.470i)13-s + (5.03 − 4.22i)14-s + (4.66 − 1.69i)16-s + (0.944 + 1.63i)17-s + (−1.37 + 2.37i)19-s + (0.161 − 0.918i)20-s + (2.95 + 1.07i)22-s + (1.01 + 5.73i)23-s + ⋯ |
| L(s) = 1 | + (−0.910 − 0.764i)2-s + (0.0716 + 0.406i)4-s + (−0.474 − 0.172i)5-s + (−0.256 + 1.45i)7-s + (−0.349 + 0.604i)8-s + (0.300 + 0.519i)10-s + (−0.529 + 0.192i)11-s + (−0.155 + 0.130i)13-s + (1.34 − 1.12i)14-s + (1.16 − 0.424i)16-s + (0.229 + 0.396i)17-s + (−0.314 + 0.544i)19-s + (0.0362 − 0.205i)20-s + (0.629 + 0.229i)22-s + (0.211 + 1.19i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.355161 + 0.220977i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.355161 + 0.220977i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (1.28 + 1.08i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (1.06 + 0.386i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.678 - 3.84i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (1.75 - 0.639i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.561 - 0.470i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.944 - 1.63i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.37 - 2.37i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.01 - 5.73i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.07 - 3.41i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.232 - 1.32i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (1.69 + 2.94i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.37 - 1.15i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (4.72 - 1.72i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.296 - 1.68i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 + (10.5 + 3.85i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.908 + 5.15i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (1.44 - 1.21i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (6.09 + 10.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.94 - 8.56i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.46 - 7.94i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.94 - 7.50i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.86 - 4.96i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.322 + 0.117i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.11204465041396024283580822223, −11.31642430904784410214084641921, −10.26216314833348603813896736810, −9.425798168419728790141277216217, −8.591708461327995859450803853875, −7.81705263724768649924559976293, −6.10917941452797856279594592822, −5.08854050437246537231941397002, −3.19344053498557107211273077074, −1.90744987158506716307283312505,
0.44297277286609786219492046567, 3.27146304982485957276178069209, 4.54047622028121367613182841715, 6.31236713689857098066005197583, 7.22503248439485818392171600851, 7.82273926589305693046517265365, 8.821865856710106179879164587875, 10.03027680135126260814130667963, 10.59169250906027462247242458472, 11.83527445363922214162976005963
