L(s) = 1 | + (0.809 + 0.587i)3-s + (−0.809 + 2.48i)5-s + (2.42 − 1.76i)7-s + (0.309 + 0.951i)9-s + (−1.69 + 2.85i)11-s + (0.545 + 1.67i)13-s + (−2.11 + 1.53i)15-s + (0.5 − 1.53i)17-s + (4.73 + 3.44i)19-s + 3·21-s − 3.47·23-s + (−1.49 − 1.08i)25-s + (−0.309 + 0.951i)27-s + (−3.61 + 2.62i)29-s + (−0.881 − 2.71i)31-s + ⋯ |
L(s) = 1 | + (0.467 + 0.339i)3-s + (−0.361 + 1.11i)5-s + (0.917 − 0.666i)7-s + (0.103 + 0.317i)9-s + (−0.509 + 0.860i)11-s + (0.151 + 0.465i)13-s + (−0.546 + 0.397i)15-s + (0.121 − 0.373i)17-s + (1.08 + 0.789i)19-s + 0.654·21-s − 0.723·23-s + (−0.299 − 0.217i)25-s + (−0.0594 + 0.183i)27-s + (−0.671 + 0.488i)29-s + (−0.158 − 0.487i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31474 + 1.00382i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31474 + 1.00382i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (1.69 - 2.85i)T \) |
good | 5 | \( 1 + (0.809 - 2.48i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-2.42 + 1.76i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-0.545 - 1.67i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 1.53i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-4.73 - 3.44i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 + (3.61 - 2.62i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.881 + 2.71i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.190 - 0.138i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-9.66 - 7.02i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + (-1.30 - 0.951i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.97 + 9.14i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-8.35 + 6.06i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.42 + 7.46i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 9.56T + 67T^{2} \) |
| 71 | \( 1 + (-1.71 + 5.29i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.61 + 1.90i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.92 + 9.00i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.218 - 0.673i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 0.527T + 89T^{2} \) |
| 97 | \( 1 + (4.33 + 13.3i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.07004932868320009130670005672, −10.12040898731993251269792462431, −9.517173080602365905039343995329, −8.023240916618786539331224873313, −7.61406963808823430139356993047, −6.73843283263469067819251960737, −5.26459553189847253737609243245, −4.21339144885266706588915706933, −3.25679001194425205530546324413, −1.90386278304849756031703084614,
1.00736001656009906182059920955, 2.50200943325781693419355793934, 3.89224251299620582378531038708, 5.14577892442151859101857090510, 5.75763102117051697649734095292, 7.36148890830955698500590822973, 8.234835579236336996756427498326, 8.605019097817071956030570618098, 9.521406257344114694775739536144, 10.81945079722963612147812222714