L(s) = 1 | − 2-s + (−1.66 − 0.468i)3-s + 4-s + (−1.58 − 0.917i)5-s + (1.66 + 0.468i)6-s + (−0.364 + 2.62i)7-s − 8-s + (2.56 + 1.56i)9-s + (1.58 + 0.917i)10-s + (1.69 − 2.93i)11-s + (−1.66 − 0.468i)12-s + (3.07 + 1.88i)13-s + (0.364 − 2.62i)14-s + (2.22 + 2.27i)15-s + 16-s − 3.18·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.962 − 0.270i)3-s + 0.5·4-s + (−0.711 − 0.410i)5-s + (0.680 + 0.191i)6-s + (−0.137 + 0.990i)7-s − 0.353·8-s + (0.853 + 0.520i)9-s + (0.502 + 0.290i)10-s + (0.511 − 0.885i)11-s + (−0.481 − 0.135i)12-s + (0.851 + 0.523i)13-s + (0.0973 − 0.700i)14-s + (0.573 + 0.587i)15-s + 0.250·16-s − 0.772·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.170165 - 0.314677i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.170165 - 0.314677i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.66 + 0.468i)T \) |
| 7 | \( 1 + (0.364 - 2.62i)T \) |
| 13 | \( 1 + (-3.07 - 1.88i)T \) |
good | 5 | \( 1 + (1.58 + 0.917i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.69 + 2.93i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 19 | \( 1 + (1.01 + 1.75i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 2.47iT - 23T^{2} \) |
| 29 | \( 1 + (1.81 - 1.04i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.14 + 5.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.503iT - 37T^{2} \) |
| 41 | \( 1 + (4.35 - 2.51i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.528 + 0.916i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (10.0 + 5.82i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.81 + 3.35i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 6.20iT - 59T^{2} \) |
| 61 | \( 1 + (-10.1 + 5.84i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.59 + 4.96i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.77 + 13.4i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.47 + 7.75i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.92 + 6.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 13.0iT - 83T^{2} \) |
| 89 | \( 1 + 5.33iT - 89T^{2} \) |
| 97 | \( 1 + (4.79 - 8.31i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.75002161471174802727154532089, −9.478078883499726772374762190045, −8.678750582486161050269732182918, −8.030157957331933952191339087953, −6.66628802482652245729887710001, −6.18556510555052517646502054161, −5.00981830701884487462789738516, −3.74195919477154755493911708371, −1.95025485171796525784368945478, −0.31774263429220653624009454155,
1.36893724573579387847364035536, 3.55074124812192134702741510846, 4.33116221873849711869634280140, 5.74658437045675542701977402349, 6.91272139819571239890051363613, 7.23144628824893104477281873790, 8.425337154530392707631002419176, 9.610010839683438896148799446796, 10.31620016851024363840151532831, 11.05322390839079279059401491546