L(s) = 1 | + (0.736 − 1.27i)2-s + (−0.0852 − 0.147i)4-s + (−1.93 + 3.34i)7-s + 2.69·8-s + (0.130 − 0.225i)11-s + (2.03 + 3.53i)13-s + (2.84 + 4.93i)14-s + (2.15 − 3.73i)16-s − 3.26·17-s + 4.24·19-s + (−0.191 − 0.332i)22-s + (4.34 + 7.53i)23-s + 6.00·26-s + 0.659·28-s + (2.11 − 3.65i)29-s + ⋯ |
L(s) = 1 | + (0.520 − 0.902i)2-s + (−0.0426 − 0.0738i)4-s + (−0.730 + 1.26i)7-s + 0.952·8-s + (0.0392 − 0.0679i)11-s + (0.565 + 0.979i)13-s + (0.761 + 1.31i)14-s + (0.538 − 0.933i)16-s − 0.790·17-s + 0.974·19-s + (−0.0408 − 0.0708i)22-s + (0.906 + 1.57i)23-s + 1.17·26-s + 0.124·28-s + (0.392 − 0.678i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.03001 + 0.0294706i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.03001 + 0.0294706i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.736 + 1.27i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (1.93 - 3.34i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.130 + 0.225i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.03 - 3.53i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + (-4.34 - 7.53i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.11 + 3.65i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.32 + 2.29i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 + (-2.82 - 4.88i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.53 + 7.85i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.714 - 1.23i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + (3.56 + 6.16i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.26 - 2.18i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.64 + 9.77i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.38T + 71T^{2} \) |
| 73 | \( 1 - 0.403T + 73T^{2} \) |
| 79 | \( 1 + (-1.52 + 2.63i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.29 + 3.96i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 7.17T + 89T^{2} \) |
| 97 | \( 1 + (1.55 - 2.69i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.84037725797185209158917642923, −9.494054166565723135587581274118, −9.183549730583586426981207623199, −7.928534812952912317186446109990, −6.86506644044213882214096738520, −5.88751370831650070596954488564, −4.84195227668476497636967519517, −3.66689312471558008549950241864, −2.83340900830698967671131051181, −1.73238277175492281038185802128,
1.00604477808104059259474071811, 3.05858940551987778118822409961, 4.19529225265420572270845069496, 5.08574041723100904842811929037, 6.18475151868277722460918035940, 6.87267198534654943980269861165, 7.52134901690764077582941448803, 8.552674085739036103412540917607, 9.720817665856905546665754918544, 10.68993929050905245467998102422