L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.806 + 1.53i)3-s + (−0.499 − 0.866i)4-s + (3.23 + 1.86i)5-s + (−1.73 − 0.0678i)6-s + (0.481 − 2.60i)7-s + 0.999·8-s + (−1.69 + 2.47i)9-s + (−3.23 + 1.86i)10-s + (−0.664 − 1.15i)11-s + (0.924 − 1.46i)12-s + (0.452 + 3.57i)13-s + (2.01 + 1.71i)14-s + (−0.253 + 6.46i)15-s + (−0.5 + 0.866i)16-s + (2.49 + 4.31i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.465 + 0.884i)3-s + (−0.249 − 0.433i)4-s + (1.44 + 0.835i)5-s + (−0.706 − 0.0276i)6-s + (0.182 − 0.983i)7-s + 0.353·8-s + (−0.566 + 0.824i)9-s + (−1.02 + 0.590i)10-s + (−0.200 − 0.346i)11-s + (0.266 − 0.422i)12-s + (0.125 + 0.992i)13-s + (0.537 + 0.459i)14-s + (−0.0653 + 1.66i)15-s + (−0.125 + 0.216i)16-s + (0.604 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03189 + 1.39990i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03189 + 1.39990i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.806 - 1.53i)T \) |
| 7 | \( 1 + (-0.481 + 2.60i)T \) |
| 13 | \( 1 + (-0.452 - 3.57i)T \) |
good | 5 | \( 1 + (-3.23 - 1.86i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.664 + 1.15i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.49 - 4.31i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.36 + 4.09i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.00 + 0.578i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6.29iT - 29T^{2} \) |
| 31 | \( 1 + (-1.66 - 2.88i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.87 + 4.54i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.38iT - 41T^{2} \) |
| 43 | \( 1 + 9.54T + 43T^{2} \) |
| 47 | \( 1 + (-6.95 - 4.01i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.36 - 0.788i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-9.31 + 5.37i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.85 - 2.80i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.51 - 2.03i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.95T + 71T^{2} \) |
| 73 | \( 1 + (7.21 + 12.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.22 + 12.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.74iT - 83T^{2} \) |
| 89 | \( 1 + (2.55 + 1.47i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 9.79T + 97T^{2} \) |
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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
−10.53140065070623609432776894702, −10.18812162123185132415402300013, −9.406892267844077614135113941540, −8.554249385239317918665013723657, −7.44841259400649329310328907076, −6.51526647354795272650413511782, −5.64760989334038432715458211057, −4.54163637817245382435682314276, −3.30242468822115084752880376978, −1.87530389118217492677664701600,
1.23329101349420309108015474954, 2.18426454973124689746246000906, 3.18004895500992527622020712358, 5.25929967676144814155151204662, 5.66913127609733097929075168070, 7.05581360278857030485012598847, 8.234011198534676387486429375259, 8.759804214177193874594411844162, 9.659993193740936022665581510140, 10.19457565508131329540811050331