| L(s) = 1 | + (0.374 − 0.158i)2-s + (−1.20 − 1.24i)3-s + (−1.27 + 1.31i)4-s + (−0.939 + 0.665i)5-s + (−0.648 − 0.272i)6-s + (−4.21 + 1.63i)7-s + (−0.562 + 1.45i)8-s + (−0.0766 + 2.99i)9-s + (−0.246 + 0.397i)10-s + (3.49 − 4.63i)11-s + (3.17 − 0.00832i)12-s + (−0.556 + 3.58i)13-s + (−1.31 + 1.27i)14-s + (1.96 + 0.361i)15-s + (−0.0935 − 3.03i)16-s + (−2.17 − 0.767i)17-s + ⋯ |
| L(s) = 1 | + (0.264 − 0.111i)2-s + (−0.698 − 0.716i)3-s + (−0.638 + 0.658i)4-s + (−0.420 + 0.297i)5-s + (−0.264 − 0.111i)6-s + (−1.59 + 0.616i)7-s + (−0.198 + 0.513i)8-s + (−0.0255 + 0.999i)9-s + (−0.0778 + 0.125i)10-s + (1.05 − 1.39i)11-s + (0.917 − 0.00240i)12-s + (−0.154 + 0.994i)13-s + (−0.352 + 0.341i)14-s + (0.506 + 0.0932i)15-s + (−0.0233 − 0.758i)16-s + (−0.527 − 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.483109 - 0.344426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.483109 - 0.344426i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.20 + 1.24i)T \) |
| 103 | \( 1 + (-9.84 + 2.44i)T \) |
| good | 2 | \( 1 + (-0.374 + 0.158i)T + (1.39 - 1.43i)T^{2} \) |
| 5 | \( 1 + (0.939 - 0.665i)T + (1.66 - 4.71i)T^{2} \) |
| 7 | \( 1 + (4.21 - 1.63i)T + (5.17 - 4.71i)T^{2} \) |
| 11 | \( 1 + (-3.49 + 4.63i)T + (-3.01 - 10.5i)T^{2} \) |
| 13 | \( 1 + (0.556 - 3.58i)T + (-12.3 - 3.94i)T^{2} \) |
| 17 | \( 1 + (2.17 + 0.767i)T + (13.2 + 10.6i)T^{2} \) |
| 19 | \( 1 + (-1.23 + 0.0761i)T + (18.8 - 2.33i)T^{2} \) |
| 23 | \( 1 + (0.693 + 5.60i)T + (-22.3 + 5.61i)T^{2} \) |
| 29 | \( 1 + (-1.58 + 3.45i)T + (-18.8 - 22.0i)T^{2} \) |
| 31 | \( 1 + (-4.86 + 0.150i)T + (30.9 - 1.90i)T^{2} \) |
| 37 | \( 1 + (-3.73 - 4.09i)T + (-3.41 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-3.06 - 0.283i)T + (40.3 + 7.53i)T^{2} \) |
| 43 | \( 1 + (3.81 + 11.9i)T + (-35.0 + 24.8i)T^{2} \) |
| 47 | \( 1 - 4.88T + 47T^{2} \) |
| 53 | \( 1 + (1.45 + 2.19i)T + (-20.6 + 48.8i)T^{2} \) |
| 59 | \( 1 + (0.158 - 0.410i)T + (-43.6 - 39.7i)T^{2} \) |
| 61 | \( 1 + (4.24 + 4.95i)T + (-9.35 + 60.2i)T^{2} \) |
| 67 | \( 1 + (14.1 - 2.19i)T + (63.8 - 20.3i)T^{2} \) |
| 71 | \( 1 + (-10.9 - 7.75i)T + (23.5 + 66.9i)T^{2} \) |
| 73 | \( 1 + (-4.98 - 0.462i)T + (71.7 + 13.4i)T^{2} \) |
| 79 | \( 1 + (-1.15 + 0.818i)T + (26.2 - 74.5i)T^{2} \) |
| 83 | \( 1 + (5.58 + 0.867i)T + (79.0 + 25.1i)T^{2} \) |
| 89 | \( 1 + (1.76 - 6.20i)T + (-75.6 - 46.8i)T^{2} \) |
| 97 | \( 1 + (-8.02 + 9.36i)T + (-14.8 - 95.8i)T^{2} \) |
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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
−9.794568425332187108216054820526, −8.953066574749529423292522455409, −8.354597949419130866520408405999, −7.10291126292524009838021932966, −6.45131633579901059782553104993, −5.79505176162654849031275813603, −4.45207162201137462255625241528, −3.49642429745909657844548279876, −2.56144105395731820415816287570, −0.39355324475608839394780899957,
0.916744256980257061555842821266, 3.34111582298013044116273435433, 4.18477922379877138519513607466, 4.75597672327292536354414651756, 5.97931872097440075756822318348, 6.49778912842307692952179153790, 7.49171836878012522273861201606, 9.006833350544561156412774562701, 9.662253657086055688574369342318, 9.997410191347628117551664495572
