L(s) = 1 | + (1.41 + 0.0966i)2-s + 3-s + (1.98 + 0.272i)4-s + 2.83i·5-s + (1.41 + 0.0966i)6-s − 0.830·7-s + (2.76 + 0.576i)8-s + 9-s + (−0.274 + 4.00i)10-s − 3.46·11-s + (1.98 + 0.272i)12-s + 0.104i·13-s + (−1.17 − 0.0802i)14-s + 2.83i·15-s + (3.85 + 1.08i)16-s + 4.98·17-s + ⋯ |
L(s) = 1 | + (0.997 + 0.0683i)2-s + 0.577·3-s + (0.990 + 0.136i)4-s + 1.27i·5-s + (0.576 + 0.0394i)6-s − 0.313·7-s + (0.979 + 0.203i)8-s + 0.333·9-s + (−0.0867 + 1.26i)10-s − 1.04·11-s + (0.571 + 0.0786i)12-s + 0.0289i·13-s + (−0.313 − 0.0214i)14-s + 0.733i·15-s + (0.962 + 0.270i)16-s + 1.20·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.00625 + 1.53781i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.00625 + 1.53781i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.41 - 0.0966i)T \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 + (-3.84 + 7.22i)T \) |
good | 5 | \( 1 - 2.83iT - 5T^{2} \) |
| 7 | \( 1 + 0.830T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 0.104iT - 13T^{2} \) |
| 17 | \( 1 - 4.98T + 17T^{2} \) |
| 19 | \( 1 - 5.61iT - 19T^{2} \) |
| 23 | \( 1 + 7.10iT - 23T^{2} \) |
| 29 | \( 1 + 0.746T + 29T^{2} \) |
| 31 | \( 1 - 2.40T + 31T^{2} \) |
| 37 | \( 1 - 0.364T + 37T^{2} \) |
| 41 | \( 1 + 9.73iT - 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 3.75iT - 47T^{2} \) |
| 53 | \( 1 - 6.44iT - 53T^{2} \) |
| 59 | \( 1 + 14.0iT - 59T^{2} \) |
| 61 | \( 1 - 8.51iT - 61T^{2} \) |
| 71 | \( 1 - 6.72iT - 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 2.89T + 79T^{2} \) |
| 83 | \( 1 + 12.2iT - 83T^{2} \) |
| 89 | \( 1 - 0.183T + 89T^{2} \) |
| 97 | \( 1 + 11.8iT - 97T^{2} \) |
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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
−10.25485886903075419746622860314, −10.09721090737108354495189322049, −8.335624851321378086870569000685, −7.67026102608788564857275181633, −6.83101457528017886155731686222, −6.05024093559703994288658230477, −5.02321699798468100888156391315, −3.67486135382804195717784665064, −3.06499965936229710707308081138, −2.12036808894156588090047681403,
1.29245347244703587821544456822, 2.72701260928392865148347242356, 3.64456989802182290831100622819, 4.93730119752406066197380545322, 5.25821490392579886250942241033, 6.52931953562635824185698569183, 7.66940252948585179928501251749, 8.210870254991386877608618585235, 9.422815015014758818449728834495, 10.01928765950891058962690787310