| L(s) = 1 | + (0.766 − 0.642i)2-s + (−1.69 − 0.340i)3-s + (0.173 − 0.984i)4-s + (−0.294 + 1.66i)5-s + (−1.51 + 0.831i)6-s + (1.80 − 1.93i)7-s + (−0.500 − 0.866i)8-s + (2.76 + 1.15i)9-s + (0.846 + 1.46i)10-s + (−0.980 − 5.56i)11-s + (−0.629 + 1.61i)12-s + (0.0358 − 0.203i)13-s + (0.142 − 2.64i)14-s + (1.06 − 2.73i)15-s + (−0.939 − 0.342i)16-s + (−1.17 − 2.03i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.980 − 0.196i)3-s + (0.0868 − 0.492i)4-s + (−0.131 + 0.745i)5-s + (−0.620 + 0.339i)6-s + (0.683 − 0.730i)7-s + (−0.176 − 0.306i)8-s + (0.922 + 0.385i)9-s + (0.267 + 0.463i)10-s + (−0.295 − 1.67i)11-s + (−0.181 + 0.465i)12-s + (0.00993 − 0.0563i)13-s + (0.0380 − 0.706i)14-s + (0.275 − 0.705i)15-s + (−0.234 − 0.0855i)16-s + (−0.285 − 0.494i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.852880 - 0.970164i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.852880 - 0.970164i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.69 + 0.340i)T \) |
| 7 | \( 1 + (-1.80 + 1.93i)T \) |
| good | 5 | \( 1 + (0.294 - 1.66i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (0.980 + 5.56i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.0358 + 0.203i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.17 + 2.03i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.71 + 2.96i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.344 + 0.289i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.32 + 7.48i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (0.450 - 2.55i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 - 2.20T + 37T^{2} \) |
| 41 | \( 1 + (1.67 - 9.50i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-5.43 + 4.55i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.59 - 9.02i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (5.33 - 9.24i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.10 + 2.22i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.548 + 3.11i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.65 - 7.26i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.0841 - 0.145i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + (11.5 - 9.65i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.953 + 5.40i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (1.50 - 2.61i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12.0 - 10.1i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.06782455090756540633897260775, −10.83380223418495269219584660293, −9.657374186668207606396549489201, −8.114820349840312049400458896915, −7.11302756377570934564713387073, −6.18838339157219850145531841409, −5.22144504807591379408608181705, −4.14678781713963460625071826597, −2.81267511974142343367714786363, −0.879433258433062068747699651222,
1.86386155323252214304205326161, 4.05710806757061964134741543376, 4.97135778622021732972028755897, 5.48752891863655693646781151895, 6.75430592538975703903363799823, 7.68817184567182696355813693030, 8.784365478325102331591926563093, 9.813842681096583771834104824028, 10.88511552101134978201753603633, 11.89637280525342468405516553144
