| L(s) = 1 | + (1.73 + 1.60i)2-s + (0.267 + 3.57i)4-s + (0.567 + 1.44i)5-s + (−2.62 + 0.356i)7-s + (−2.32 + 2.91i)8-s + (−1.34 + 3.41i)10-s + (0.663 − 0.204i)11-s + (−0.928 + 4.06i)13-s + (−5.11 − 3.59i)14-s + (−1.64 + 0.247i)16-s + (0.659 + 0.449i)17-s + (1.05 − 1.82i)19-s + (−5.01 + 2.41i)20-s + (1.47 + 0.711i)22-s + (4.52 − 3.08i)23-s + ⋯ |
| L(s) = 1 | + (1.22 + 1.13i)2-s + (0.133 + 1.78i)4-s + (0.253 + 0.647i)5-s + (−0.990 + 0.134i)7-s + (−0.823 + 1.03i)8-s + (−0.424 + 1.08i)10-s + (0.200 − 0.0616i)11-s + (−0.257 + 1.12i)13-s + (−1.36 − 0.960i)14-s + (−0.410 + 0.0618i)16-s + (0.159 + 0.109i)17-s + (0.241 − 0.418i)19-s + (−1.12 + 0.540i)20-s + (0.314 + 0.151i)22-s + (0.944 − 0.643i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.984118 + 2.26564i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.984118 + 2.26564i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.62 - 0.356i)T \) |
| good | 2 | \( 1 + (-1.73 - 1.60i)T + (0.149 + 1.99i)T^{2} \) |
| 5 | \( 1 + (-0.567 - 1.44i)T + (-3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.663 + 0.204i)T + (9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (0.928 - 4.06i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.659 - 0.449i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (-1.05 + 1.82i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.52 + 3.08i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (0.0358 - 0.0172i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-1.21 - 2.10i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.720 + 9.60i)T + (-36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (2.97 - 3.73i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (4.04 + 5.07i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (4.10 + 3.81i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.686 - 9.15i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-3.92 + 10.0i)T + (-43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (0.356 - 4.75i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-3.95 - 6.85i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.10 - 1.49i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (1.84 - 1.71i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (-5.31 + 9.20i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.05 + 9.01i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-0.0252 - 0.00779i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−11.79745566689470758031041659195, −10.61365442151745786633404115758, −9.539452315894267362112649455063, −8.559387122654582576128459644270, −7.07894186184451323715118823239, −6.80511727388897865876390880774, −5.93587431264971508379444333103, −4.84095925668697448267081165246, −3.74316734568209079019996793797, −2.70056365134293051764751639685,
1.19193282233571685511181519117, 2.84557578097585457254758982674, 3.60656737041409539388311629266, 4.91420098726580002933176284918, 5.57425549261905244650068115127, 6.70508352933001038545065888215, 8.125938966857046327785211544494, 9.497078777040244556122754629973, 10.01641846926878876154358488283, 10.97205813885133372590825211708
