L(s) = 1 | + (−0.0747 + 0.997i)2-s + (0.563 − 0.173i)3-s + (−0.988 − 0.149i)4-s + (0.131 + 0.574i)6-s + (0.974 + 0.222i)7-s + (0.222 − 0.974i)8-s + (−0.539 + 0.367i)9-s + (1.64 + 1.12i)11-s + (−0.582 + 0.0878i)12-s + (0.134 − 0.0648i)13-s + (−0.294 + 0.955i)14-s + (0.955 + 0.294i)16-s + (0.365 − 0.930i)17-s + (−0.326 − 0.565i)18-s + (0.587 − 0.0440i)21-s + (−1.24 + 1.55i)22-s + ⋯ |
L(s) = 1 | + (−0.0747 + 0.997i)2-s + (0.563 − 0.173i)3-s + (−0.988 − 0.149i)4-s + (0.131 + 0.574i)6-s + (0.974 + 0.222i)7-s + (0.222 − 0.974i)8-s + (−0.539 + 0.367i)9-s + (1.64 + 1.12i)11-s + (−0.582 + 0.0878i)12-s + (0.134 − 0.0648i)13-s + (−0.294 + 0.955i)14-s + (0.955 + 0.294i)16-s + (0.365 − 0.930i)17-s + (−0.326 − 0.565i)18-s + (0.587 − 0.0440i)21-s + (−1.24 + 1.55i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.559813590\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.559813590\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0747 - 0.997i)T \) |
| 7 | \( 1 + (-0.974 - 0.222i)T \) |
| 17 | \( 1 + (-0.365 + 0.930i)T \) |
good | 3 | \( 1 + (-0.563 + 0.173i)T + (0.826 - 0.563i)T^{2} \) |
| 5 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 11 | \( 1 + (-1.64 - 1.12i)T + (0.365 + 0.930i)T^{2} \) |
| 13 | \( 1 + (-0.134 + 0.0648i)T + (0.623 - 0.781i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.317 + 0.807i)T + (-0.733 + 0.680i)T^{2} \) |
| 29 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (0.974 + 1.68i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 41 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 53 | \( 1 + (-1.44 - 0.218i)T + (0.955 + 0.294i)T^{2} \) |
| 59 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 61 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.848 - 1.06i)T + (-0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 79 | \( 1 + (0.563 - 0.975i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (1.57 - 1.07i)T + (0.365 - 0.930i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.879228673527052182604219337966, −8.191649188750002291752937344072, −7.38893791859686156573699503159, −7.06587385162726456630109650342, −5.94694402924732810504768892924, −5.29855395690687887493890130888, −4.42513214702375879433230474366, −3.77851012184916587119019539336, −2.39489153686810657834653619247, −1.36233289576536783533925602750,
1.11637839363043069283824711332, 1.93159265085208418909032408416, 3.24718549733350613309028212427, 3.70035151841051803412871416365, 4.41603966073299315516030353810, 5.54198496784861029943458864762, 6.20051170072508353142062040719, 7.40167221607972367614817489777, 8.441487604820978061517895637737, 8.619116031453930910147080384323