| L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.110 − 0.767i)5-s + (−0.841 − 0.540i)6-s + (0.415 − 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.322 + 0.705i)10-s + (−1.03 − 0.302i)11-s + (0.959 + 0.281i)12-s + (−1.81 − 3.96i)13-s + (−0.142 + 0.989i)14-s + (0.507 − 0.586i)15-s + (0.415 − 0.909i)16-s + (−4.59 − 2.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (−0.0493 − 0.343i)5-s + (−0.343 − 0.220i)6-s + (0.157 − 0.343i)7-s + (−0.231 + 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.101 + 0.223i)10-s + (−0.310 − 0.0911i)11-s + (0.276 + 0.0813i)12-s + (−0.502 − 1.09i)13-s + (−0.0380 + 0.264i)14-s + (0.131 − 0.151i)15-s + (0.103 − 0.227i)16-s + (−1.11 − 0.715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0265 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0265 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.587788 - 0.572355i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.587788 - 0.572355i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (3.31 + 3.46i)T \) |
| good | 5 | \( 1 + (0.110 + 0.767i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (1.03 + 0.302i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (1.81 + 3.96i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (4.59 + 2.95i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.993 + 0.638i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-2.11 - 1.36i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (3.19 - 3.69i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.536 + 3.73i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.392 + 2.73i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (7.23 + 8.34i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 8.68T + 47T^{2} \) |
| 53 | \( 1 + (1.43 - 3.14i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-2.35 - 5.16i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (1.16 - 1.34i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-13.0 + 3.83i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-2.97 + 0.873i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-12.3 + 7.94i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (2.51 + 5.51i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (0.124 - 0.866i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (4.75 + 5.48i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (1.00 + 6.99i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−9.806395290093381796514795300842, −8.857659882039511746701698379940, −8.335456516636769492108938547328, −7.44819636799033449853572111690, −6.63978228447914806926324231422, −5.32441019626545539290711169120, −4.66482980151135903244363440658, −3.29136902410611948059324653194, −2.19833320788941959185341443441, −0.43213085349459053680955285820,
1.67293559579068566780791625564, 2.51251522190478824534354773290, 3.72026747760018099307186122489, 4.94458817282766685625657702415, 6.36335450684219062865215390441, 6.86997973128594084472738824277, 7.924362482321957759383465453758, 8.467497484121893782933152593831, 9.459996781164585921527012506381, 9.960064739705850073519150473094
