L(s) = 1 | − 2-s + (1.67 + 0.423i)3-s + 4-s + i·5-s + (−1.67 − 0.423i)6-s − 2.47·7-s − 8-s + (2.64 + 1.42i)9-s − i·10-s + 2i·11-s + (1.67 + 0.423i)12-s − 1.15i·13-s + 2.47·14-s + (−0.423 + 1.67i)15-s + 16-s + 6.67i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.969 + 0.244i)3-s + 0.5·4-s + 0.447i·5-s + (−0.685 − 0.173i)6-s − 0.934·7-s − 0.353·8-s + (0.880 + 0.474i)9-s − 0.316i·10-s + 0.603i·11-s + (0.484 + 0.122i)12-s − 0.319i·13-s + 0.660·14-s + (−0.109 + 0.433i)15-s + 0.250·16-s + 1.61i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01658 + 0.799147i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01658 + 0.799147i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.67 - 0.423i)T \) |
| 5 | \( 1 - iT \) |
| 19 | \( 1 + (-4.35 + 0.0388i)T \) |
good | 7 | \( 1 + 2.47T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 1.15iT - 13T^{2} \) |
| 17 | \( 1 - 6.67iT - 17T^{2} \) |
| 23 | \( 1 - 5.79iT - 23T^{2} \) |
| 29 | \( 1 - 3.35T + 29T^{2} \) |
| 31 | \( 1 - 4.80iT - 31T^{2} \) |
| 37 | \( 1 + 11.6iT - 37T^{2} \) |
| 41 | \( 1 + 7.83T + 41T^{2} \) |
| 43 | \( 1 + 0.978T + 43T^{2} \) |
| 47 | \( 1 + 3.02iT - 47T^{2} \) |
| 53 | \( 1 + 2.33T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 10.0iT - 67T^{2} \) |
| 71 | \( 1 + 7.24T + 71T^{2} \) |
| 73 | \( 1 + 6.97T + 73T^{2} \) |
| 79 | \( 1 + 5.75iT - 79T^{2} \) |
| 83 | \( 1 + 11.1iT - 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 6.94iT - 97T^{2} \) |
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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
−10.36148514433256066588034717825, −10.09914619002337600874600486332, −9.214557706365913457552199240776, −8.388294725079460078929974321868, −7.44586988872774273092883006887, −6.77647872772529472599620100914, −5.52830050211903046114216534010, −3.84585071863623973304417637436, −3.08358704447890703485588413940, −1.77745742364231431421247865690,
0.856071556048657012534188796177, 2.56196079785262663743072419009, 3.40745057945928375366645701001, 4.88275114047376043511820139011, 6.39014469209090615738762997765, 7.07229827817652047091864808636, 8.117132046056452857661158306095, 8.794553809677792666895670993225, 9.645659916593142150890156168612, 10.03846031916087198525421902819