| L(s) = 1 | + (−0.995 − 0.0950i)2-s + (−0.690 + 0.723i)3-s + (0.981 + 0.189i)4-s + (2.93 − 1.51i)5-s + (0.755 − 0.654i)6-s + (−2.06 − 1.65i)7-s + (−0.959 − 0.281i)8-s + (−0.0475 − 0.998i)9-s + (−3.06 + 1.22i)10-s + (0.0879 + 0.920i)11-s + (−0.814 + 0.580i)12-s + (5.70 + 0.819i)13-s + (1.89 + 1.84i)14-s + (−0.930 + 3.17i)15-s + (0.928 + 0.371i)16-s + (−0.574 + 1.65i)17-s + ⋯ |
| L(s) = 1 | + (−0.703 − 0.0672i)2-s + (−0.398 + 0.417i)3-s + (0.490 + 0.0946i)4-s + (1.31 − 0.677i)5-s + (0.308 − 0.267i)6-s + (−0.780 − 0.625i)7-s + (−0.339 − 0.0996i)8-s + (−0.0158 − 0.332i)9-s + (−0.970 + 0.388i)10-s + (0.0265 + 0.277i)11-s + (−0.235 + 0.167i)12-s + (1.58 + 0.227i)13-s + (0.507 + 0.492i)14-s + (−0.240 + 0.818i)15-s + (0.232 + 0.0929i)16-s + (−0.139 + 0.402i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07882 - 0.473077i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07882 - 0.473077i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.995 + 0.0950i)T \) |
| 3 | \( 1 + (0.690 - 0.723i)T \) |
| 7 | \( 1 + (2.06 + 1.65i)T \) |
| 23 | \( 1 + (-0.861 + 4.71i)T \) |
| good | 5 | \( 1 + (-2.93 + 1.51i)T + (2.90 - 4.07i)T^{2} \) |
| 11 | \( 1 + (-0.0879 - 0.920i)T + (-10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (-5.70 - 0.819i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.574 - 1.65i)T + (-13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (0.560 + 1.62i)T + (-14.9 + 11.7i)T^{2} \) |
| 29 | \( 1 + (2.01 + 2.32i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (2.36 - 0.574i)T + (27.5 - 14.2i)T^{2} \) |
| 37 | \( 1 + (-1.39 + 0.0662i)T + (36.8 - 3.51i)T^{2} \) |
| 41 | \( 1 + (-6.30 + 9.80i)T + (-17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-0.343 - 1.16i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (-3.87 - 2.23i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.80 + 4.83i)T + (-12.4 + 51.5i)T^{2} \) |
| 59 | \( 1 + (-0.215 - 0.537i)T + (-42.7 + 40.7i)T^{2} \) |
| 61 | \( 1 + (1.79 - 1.71i)T + (2.90 - 60.9i)T^{2} \) |
| 67 | \( 1 + (-4.09 - 2.91i)T + (21.9 + 63.3i)T^{2} \) |
| 71 | \( 1 + (6.63 + 14.5i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (0.470 - 2.44i)T + (-67.7 - 27.1i)T^{2} \) |
| 79 | \( 1 + (-6.16 + 7.83i)T + (-18.6 - 76.7i)T^{2} \) |
| 83 | \( 1 + (-2.78 + 1.78i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-2.83 + 11.6i)T + (-79.1 - 40.7i)T^{2} \) |
| 97 | \( 1 + (-10.8 - 6.94i)T + (40.2 + 88.2i)T^{2} \) |
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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
−9.886132189423538831560444279210, −9.106425737186384607763556102587, −8.706112406569824677722159271685, −7.33528207219891173815236069978, −6.22129756068268182463527273211, −5.98549590315022730289581344443, −4.62813676644235201092634125680, −3.56549896684365170498759225600, −2.05738533879030940041074996124, −0.810827573840507911688724130578,
1.31154448978557655232293456545, 2.45834992176258259908324922539, 3.45944317594592028150593557476, 5.47619886127900475535298560126, 6.06245050121572238338725363475, 6.51894734689223099279822308917, 7.53487974762035071166643537508, 8.648602290126382540296252322388, 9.367887181339954523923407643693, 10.01383105663376695039069158152
