| L(s) = 1 | + (1.62 + 0.586i)3-s + (1.87 − 0.684i)4-s + (−1.43 + 1.71i)5-s + (2.48 + 0.904i)7-s + (2.31 + 1.91i)9-s + (−1.23 − 1.47i)11-s + (3.46 − 0.0122i)12-s + (−0.396 + 2.25i)13-s + (−3.34 + 1.94i)15-s + (3.06 − 2.57i)16-s + (0.531 + 0.306i)17-s + (−1.52 + 4.20i)20-s + (3.52 + 2.93i)21-s + (−0.868 − 4.92i)25-s + (2.64 + 4.47i)27-s + 5.29·28-s + ⋯ |
| L(s) = 1 | + (0.940 + 0.338i)3-s + (0.939 − 0.342i)4-s + (−0.642 + 0.766i)5-s + (0.939 + 0.342i)7-s + (0.770 + 0.637i)9-s + (−0.373 − 0.445i)11-s + (0.999 − 0.00354i)12-s + (−0.110 + 0.624i)13-s + (−0.864 + 0.503i)15-s + (0.766 − 0.642i)16-s + (0.128 + 0.0743i)17-s + (−0.342 + 0.939i)20-s + (0.768 + 0.640i)21-s + (−0.173 − 0.984i)25-s + (0.509 + 0.860i)27-s + 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.47430 + 0.998235i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.47430 + 0.998235i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.62 - 0.586i)T \) |
| 5 | \( 1 + (1.43 - 1.71i)T \) |
| 7 | \( 1 + (-2.48 - 0.904i)T \) |
| good | 2 | \( 1 + (-1.87 + 0.684i)T^{2} \) |
| 11 | \( 1 + (1.23 + 1.47i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.396 - 2.25i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.531 - 0.306i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (4.76 - 0.839i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.18 + 3.25i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (13.1 + 7.61i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.67 - 4.63i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.06 + 17.3i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-16.9 + 2.98i)T + (77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (15.0 - 12.6i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.34667252381090290049308401100, −9.286429913145462299750125275120, −8.328755269360381255198384973553, −7.64049181285435774822199963206, −7.02725929240956031929116869388, −5.88653013155315014739046970771, −4.76150945759975132055770042156, −3.63150283606479537284178731908, −2.68035660535463037620077723620, −1.78411939051112212754820617222,
1.26964127122359989172012901206, 2.35223712681213826899905618858, 3.51317233556224155135673126056, 4.42703220284342011572253916535, 5.55353348817606009262201414877, 6.95567087696766769844865103238, 7.66885696220447474618683730077, 8.014747670280903097687387460922, 8.836821777138933691409310526011, 9.911924578835939100208135530852
