| 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
−11.27242885865118615142460422518, −10.54301411925851508192273972313, −9.527506744702116432769693412691, −8.767784745507303180977048014036, −7.37320508124977632708245759572, −5.91332293936971932181898323339, −5.11038662522268569359850159018, −4.64974438825815517332957257324, −3.02647584103625822243227247591, −1.15270034667639203422080359598,
1.68027861238686062355452442399, 3.55508272619930185320590935019, 4.71872323844087867142801414899, 5.94349870200600329500266007227, 6.62705481398836197521207708130, 7.43315803305889633644316443458, 8.346244267750668492601930575827, 9.877024058413329805364380849177, 11.02588004029560789453090177378, 11.42323067247627192936853230964
