| L(s) = 1 | + (0.766 + 0.642i)2-s + (−1.42 − 0.981i)3-s + (0.173 + 0.984i)4-s + (0.343 + 1.94i)5-s + (−0.462 − 1.66i)6-s + (1.84 − 1.89i)7-s + (−0.500 + 0.866i)8-s + (1.07 + 2.80i)9-s + (−0.989 + 1.71i)10-s + (0.568 − 3.22i)11-s + (0.718 − 1.57i)12-s + (1.17 + 6.64i)13-s + (2.63 − 0.270i)14-s + (1.42 − 3.11i)15-s + (−0.939 + 0.342i)16-s + (0.298 − 0.517i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (−0.824 − 0.566i)3-s + (0.0868 + 0.492i)4-s + (0.153 + 0.871i)5-s + (−0.188 − 0.681i)6-s + (0.696 − 0.717i)7-s + (−0.176 + 0.306i)8-s + (0.358 + 0.933i)9-s + (−0.312 + 0.541i)10-s + (0.171 − 0.972i)11-s + (0.207 − 0.454i)12-s + (0.324 + 1.84i)13-s + (0.703 − 0.0722i)14-s + (0.367 − 0.805i)15-s + (−0.234 + 0.0855i)16-s + (0.0724 − 0.125i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38851 + 0.687136i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38851 + 0.687136i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.42 + 0.981i)T \) |
| 7 | \( 1 + (-1.84 + 1.89i)T \) |
| good | 5 | \( 1 + (-0.343 - 1.94i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.568 + 3.22i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.17 - 6.64i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.298 + 0.517i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.15 - 7.20i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.380 + 0.319i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.221 + 1.25i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.261 + 1.48i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + 0.542T + 37T^{2} \) |
| 41 | \( 1 + (-0.421 - 2.39i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (2.94 + 2.47i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (0.157 - 0.894i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (5.09 + 8.82i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (12.6 + 4.60i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.27 + 7.23i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (2.08 - 1.75i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.32 + 2.29i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 1.35T + 73T^{2} \) |
| 79 | \( 1 + (-5.44 - 4.56i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.31 - 7.44i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (8.63 + 14.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.38 + 5.35i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.42323067247627192936853230964, −11.02588004029560789453090177378, −9.877024058413329805364380849177, −8.346244267750668492601930575827, −7.43315803305889633644316443458, −6.62705481398836197521207708130, −5.94349870200600329500266007227, −4.71872323844087867142801414899, −3.55508272619930185320590935019, −1.68027861238686062355452442399,
1.15270034667639203422080359598, 3.02647584103625822243227247591, 4.64974438825815517332957257324, 5.11038662522268569359850159018, 5.91332293936971932181898323339, 7.37320508124977632708245759572, 8.767784745507303180977048014036, 9.527506744702116432769693412691, 10.54301411925851508192273972313, 11.27242885865118615142460422518
