| L(s) = 1 | + (0.415 − 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.698 + 0.449i)5-s + (−0.654 + 0.755i)6-s + (0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (0.118 + 0.822i)10-s + (1.99 + 4.36i)11-s + (0.415 + 0.909i)12-s + (0.352 + 2.45i)13-s + (−0.841 − 0.540i)14-s + (0.797 − 0.234i)15-s + (−0.142 + 0.989i)16-s + (3.25 − 3.75i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (−0.553 − 0.162i)3-s + (−0.327 − 0.377i)4-s + (−0.312 + 0.200i)5-s + (−0.267 + 0.308i)6-s + (0.0537 − 0.374i)7-s + (−0.339 + 0.0996i)8-s + (0.280 + 0.180i)9-s + (0.0373 + 0.260i)10-s + (0.600 + 1.31i)11-s + (0.119 + 0.262i)12-s + (0.0978 + 0.680i)13-s + (−0.224 − 0.144i)14-s + (0.205 − 0.0604i)15-s + (−0.0355 + 0.247i)16-s + (0.790 − 0.911i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37889 - 0.431113i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37889 - 0.431113i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 3 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (-2.91 - 3.80i)T \) |
| good | 5 | \( 1 + (0.698 - 0.449i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (-1.99 - 4.36i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.352 - 2.45i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-3.25 + 3.75i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (1.64 + 1.89i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-3.09 + 3.56i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-7.27 + 2.13i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-4.15 - 2.67i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-0.216 + 0.139i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (7.79 + 2.28i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + (-1.71 + 11.9i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (1.34 + 9.33i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-9.30 + 2.73i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (2.99 - 6.56i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (4.36 - 9.56i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-5.86 - 6.77i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-2.36 - 16.4i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-7.23 - 4.65i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-4.51 - 1.32i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-2.23 + 1.43i)T + (40.2 - 88.2i)T^{2} \) |
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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
−9.737888672624852323702021123727, −9.693847089563845485514088884825, −8.248858420198718499392760096538, −7.14545941468813032030216594550, −6.68978956193222313002164529939, −5.36888608622069013431908833469, −4.55025732683226675191390651357, −3.73809787612452866644689820341, −2.35389833545966526448040399605, −1.06327059071512895174071070592,
0.890762265607183411131420896285, 3.00941226612840796523312434664, 4.00369424902490937392770621106, 4.95451154067324726017447587391, 6.07379958608927852497870376052, 6.22508450399404469879439759210, 7.62982811678861143718640219300, 8.424432779032895384263867435649, 8.942060953130796656309995789212, 10.28693165581253403564406555972
