| L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.402 − 1.68i)3-s + (0.173 + 0.984i)4-s + (−0.366 − 2.08i)5-s + (0.774 − 1.54i)6-s + (1.84 − 1.89i)7-s + (−0.500 + 0.866i)8-s + (−2.67 + 1.35i)9-s + (1.05 − 1.83i)10-s + (−0.394 + 2.23i)11-s + (1.58 − 0.689i)12-s + (−0.774 − 4.39i)13-s + (2.63 − 0.264i)14-s + (−3.35 + 1.45i)15-s + (−0.939 + 0.342i)16-s + (0.550 − 0.953i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (−0.232 − 0.972i)3-s + (0.0868 + 0.492i)4-s + (−0.164 − 0.930i)5-s + (0.316 − 0.632i)6-s + (0.697 − 0.716i)7-s + (−0.176 + 0.306i)8-s + (−0.891 + 0.452i)9-s + (0.334 − 0.578i)10-s + (−0.119 + 0.675i)11-s + (0.458 − 0.198i)12-s + (−0.214 − 1.21i)13-s + (0.703 − 0.0707i)14-s + (−0.867 + 0.376i)15-s + (−0.234 + 0.0855i)16-s + (0.133 − 0.231i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.282 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26632 - 0.947646i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26632 - 0.947646i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 + (0.402 + 1.68i)T \) |
| 7 | \( 1 + (-1.84 + 1.89i)T \) |
| good | 5 | \( 1 + (0.366 + 2.08i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (0.394 - 2.23i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (0.774 + 4.39i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.550 + 0.953i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.27 + 5.67i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.32 + 3.63i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.885 - 5.02i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.23 - 6.97i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + (-1.28 - 7.30i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (4.90 + 4.11i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (1.25 - 7.11i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-2.35 - 4.07i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.22 + 1.17i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.248 - 1.40i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (5.47 - 4.58i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.73 + 4.74i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + (-4.38 - 3.68i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.173 - 0.985i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-4.05 - 7.03i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.607 - 0.509i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.34240741858987302249908324701, −10.56942081353909697863024421208, −8.939151456565562265030030371867, −8.119091004682051864784872580058, −7.37698307009794733458760289951, −6.51512283807072836494627683026, −5.06553397609364208404599123143, −4.70050772887689392573413926702, −2.77823318853889964975019292280, −0.980216762950352103460008900914,
2.27621970657273414864496020494, 3.52350758230390491585014947343, 4.46622200240916317047655902250, 5.64682035604093228944074588421, 6.39277771774005095069203959101, 7.924793759840118746200425247820, 9.048248786768089419371403225341, 9.924656045523324318206807029439, 10.92237817873244738647365641283, 11.41161427904475291420220018201
