L(s) = 1 | + (−0.222 + 0.0250i)2-s + (0.781 − 0.623i)3-s + (−0.926 + 0.211i)4-s + (−0.158 + 0.158i)6-s + (0.412 − 0.144i)8-s + (0.222 − 0.974i)9-s + (−0.592 + 0.742i)12-s + (0.541 + 1.12i)13-s + (0.767 − 0.369i)16-s + (−0.0250 + 0.222i)18-s + (0.781 − 0.623i)23-s + (0.232 − 0.369i)24-s + (−0.222 − 0.974i)25-s + (−0.148 − 0.236i)26-s + (−0.433 − 0.900i)27-s + ⋯ |
L(s) = 1 | + (−0.222 + 0.0250i)2-s + (0.781 − 0.623i)3-s + (−0.926 + 0.211i)4-s + (−0.158 + 0.158i)6-s + (0.412 − 0.144i)8-s + (0.222 − 0.974i)9-s + (−0.592 + 0.742i)12-s + (0.541 + 1.12i)13-s + (0.767 − 0.369i)16-s + (−0.0250 + 0.222i)18-s + (0.781 − 0.623i)23-s + (0.232 − 0.369i)24-s + (−0.222 − 0.974i)25-s + (−0.148 − 0.236i)26-s + (−0.433 − 0.900i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.149143085\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.149143085\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.781 + 0.623i)T \) |
| 23 | \( 1 + (-0.781 + 0.623i)T \) |
| 29 | \( 1 + (-0.974 + 0.222i)T \) |
good | 2 | \( 1 + (0.222 - 0.0250i)T + (0.974 - 0.222i)T^{2} \) |
| 5 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 7 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 11 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 13 | \( 1 + (-0.541 - 1.12i)T + (-0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 31 | \( 1 + (-0.158 - 1.40i)T + (-0.974 + 0.222i)T^{2} \) |
| 37 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 41 | \( 1 + (-0.467 + 0.467i)T - iT^{2} \) |
| 43 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 47 | \( 1 + (-0.559 + 1.59i)T + (-0.781 - 0.623i)T^{2} \) |
| 53 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 59 | \( 1 + 0.445iT - T^{2} \) |
| 61 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 67 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 71 | \( 1 + (-1.40 + 0.678i)T + (0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (0.189 - 1.68i)T + (-0.974 - 0.222i)T^{2} \) |
| 79 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 83 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 97 | \( 1 + (0.433 + 0.900i)T^{2} \) |
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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
−8.957768278728986689007762566344, −8.609922920377956105440096908285, −7.956459620547753135648399131319, −6.93355952090672692024948494010, −6.43840925313676481590651129314, −5.10545759767552508776844115708, −4.22043709217125762580017042173, −3.46443150750175638704438592816, −2.32180074601726344856533534853, −1.05559255762933578757781712889,
1.28022904199962708516313214069, 2.81856833613691042997145679893, 3.62152374103860928846360416954, 4.50244551083679416496164384324, 5.25774154126959543653108378247, 6.06062776537738905821480871039, 7.53125185520391169211932611928, 7.991354584225347152279755980577, 8.765316433327222234366156878150, 9.419793751861336711981719706396