| L(s) = 1 | + (−1.10 − 2.41i)2-s + (−0.959 + 0.281i)3-s + (−3.32 + 3.83i)4-s + (−2.44 − 1.57i)5-s + (1.74 + 2.01i)6-s + (−0.326 − 2.27i)7-s + (7.84 + 2.30i)8-s + (0.841 − 0.540i)9-s + (−1.09 + 7.65i)10-s + (0.0494 − 0.108i)11-s + (2.10 − 4.61i)12-s + (0.614 − 4.27i)13-s + (−5.13 + 3.30i)14-s + (2.78 + 0.818i)15-s + (−1.65 − 11.4i)16-s + (−1.68 − 1.94i)17-s + ⋯ |
| L(s) = 1 | + (−0.781 − 1.71i)2-s + (−0.553 + 0.162i)3-s + (−1.66 + 1.91i)4-s + (−1.09 − 0.702i)5-s + (0.711 + 0.820i)6-s + (−0.123 − 0.859i)7-s + (2.77 + 0.814i)8-s + (0.280 − 0.180i)9-s + (−0.347 + 2.41i)10-s + (0.0149 − 0.0326i)11-s + (0.608 − 1.33i)12-s + (0.170 − 1.18i)13-s + (−1.37 + 0.882i)14-s + (0.719 + 0.211i)15-s + (−0.412 − 2.87i)16-s + (−0.408 − 0.471i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0538963 + 0.333582i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0538963 + 0.333582i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (-4.00 - 2.63i)T \) |
| good | 2 | \( 1 + (1.10 + 2.41i)T + (-1.30 + 1.51i)T^{2} \) |
| 5 | \( 1 + (2.44 + 1.57i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (0.326 + 2.27i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.0494 + 0.108i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.614 + 4.27i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (1.68 + 1.94i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (1.13 - 1.30i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (0.933 + 1.07i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-5.49 - 1.61i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-5.79 + 3.72i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (5.48 + 3.52i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (8.28 - 2.43i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 5.91T + 47T^{2} \) |
| 53 | \( 1 + (0.514 + 3.58i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.386 + 2.68i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (12.0 + 3.54i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (0.0349 + 0.0766i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-6.43 - 14.0i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-0.455 + 0.525i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.05 + 7.31i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-2.33 + 1.50i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-12.7 + 3.74i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (7.16 + 4.60i)T + (40.2 + 88.2i)T^{2} \) |
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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
−13.45046162063465024581903964176, −12.62834240555599486661928678332, −11.70204301661528940804647839516, −10.87935133475813901846991300894, −9.953764795004753039716353727712, −8.617126654641175566688755562502, −7.57784457935223852435482549628, −4.67498834534345775658407947491, −3.50005111442000276190688462611, −0.66629912167542182948984053402,
4.61822380891954624331215452690, 6.24513806444562262689110926053, 6.97416780929499726909304581069, 8.190685769740301194958174095883, 9.222221872204522288935265088599, 10.72827771769509947229783244937, 11.90377938968463380229688613476, 13.58477660852123891430460098387, 15.04677012416509183050885681345, 15.26949735471462060456014250718
