| L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.906 + 1.98i)5-s + (−0.142 + 0.989i)6-s + (0.959 − 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (2.09 + 0.614i)10-s + (1.84 + 2.12i)11-s + (0.654 + 0.755i)12-s + (−1.69 − 0.496i)13-s + (0.415 − 0.909i)14-s + (−1.83 − 1.18i)15-s + (−0.959 + 0.281i)16-s + (−0.0157 + 0.109i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.405 + 0.887i)5-s + (−0.0580 + 0.404i)6-s + (0.362 − 0.106i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (0.662 + 0.194i)10-s + (0.555 + 0.641i)11-s + (0.189 + 0.218i)12-s + (−0.468 − 0.137i)13-s + (0.111 − 0.243i)14-s + (−0.474 − 0.304i)15-s + (−0.239 + 0.0704i)16-s + (−0.00383 + 0.0266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.81178 + 0.460643i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81178 + 0.460643i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 3 | \( 1 + (0.841 - 0.540i)T \) |
| 7 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (-3.81 + 2.91i)T \) |
| good | 5 | \( 1 + (-0.906 - 1.98i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (-1.84 - 2.12i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.69 + 0.496i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.0157 - 0.109i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.891 - 6.19i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (1.31 - 9.17i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (0.277 + 0.178i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.55 + 3.39i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-4.33 - 9.50i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (1.16 - 0.746i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 + (-7.12 + 2.09i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-8.70 - 2.55i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-7.83 - 5.03i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (1.11 - 1.28i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-10.7 + 12.4i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-2.04 - 14.2i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (13.8 + 4.08i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-5.50 + 12.0i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (6.57 - 4.22i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (2.55 + 5.59i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.15712943004796380600652658535, −9.715949584921210102507130716488, −8.566377281917046682999847627740, −7.26608542861523808345221590398, −6.59409584809495027106613518164, −5.63888064277070682092621664410, −4.77702067023580254247369873327, −3.79961468999830883316152072256, −2.72863182506208380425448747268, −1.46827958852224271017553589703,
0.881600678960295323212659339049, 2.41181162069937208058349691241, 3.93713444061883383020117968648, 5.01572563199105179299272707667, 5.44639113733397986485683413853, 6.48822482079967197352583084673, 7.24777768205971025370889766013, 8.244899718144809019708544694123, 9.030302281518091595185403991317, 9.711335490805079568317900843433
