L(s) = 1 | + (1.05 − 0.946i)2-s + (−0.913 − 0.406i)3-s + (0.104 − 0.994i)4-s + (−0.669 + 0.743i)5-s + (−1.34 + 0.437i)6-s + (0.575 + 1.29i)7-s + (0.669 + 0.743i)9-s + 1.41i·10-s + (−0.500 + 0.866i)12-s + (1.82 + 0.813i)14-s + (0.913 − 0.406i)15-s + (0.978 + 0.207i)16-s + (1.40 + 0.147i)18-s + (0.669 + 0.743i)20-s − 1.41i·21-s + ⋯ |
L(s) = 1 | + (1.05 − 0.946i)2-s + (−0.913 − 0.406i)3-s + (0.104 − 0.994i)4-s + (−0.669 + 0.743i)5-s + (−1.34 + 0.437i)6-s + (0.575 + 1.29i)7-s + (0.669 + 0.743i)9-s + 1.41i·10-s + (−0.500 + 0.866i)12-s + (1.82 + 0.813i)14-s + (0.913 − 0.406i)15-s + (0.978 + 0.207i)16-s + (1.40 + 0.147i)18-s + (0.669 + 0.743i)20-s − 1.41i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.284773695\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.284773695\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.913 + 0.406i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.05 + 0.946i)T + (0.104 - 0.994i)T^{2} \) |
| 5 | \( 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2} \) |
| 7 | \( 1 + (-0.575 - 1.29i)T + (-0.669 + 0.743i)T^{2} \) |
| 13 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.575 + 1.29i)T + (-0.669 + 0.743i)T^{2} \) |
| 31 | \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
| 61 | \( 1 + (-0.294 + 1.38i)T + (-0.913 - 0.406i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 83 | \( 1 + (-0.294 + 1.38i)T + (-0.913 - 0.406i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
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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
−10.59563603190611403219494236229, −9.480441622796539108296863935436, −8.164046315146408776710341509141, −7.51455512731715430167370550844, −6.32483036067485583892294248528, −5.56755955973650092693467775722, −4.83200308787896975862986747301, −3.84794538425878646703534785236, −2.70220341436518213873675353123, −1.80493254958784447169420725039,
1.06388463133852421258193741947, 3.79030993878787212130996749754, 4.14120730302850729160900735349, 4.99514543658973737994916016135, 5.57091167949365381186116863995, 6.72954713394311842050425191494, 7.31391134954293686158908046456, 8.067494888170768481077183468496, 9.252189935972161604338638168278, 10.33490349711704467970356567532