| L(s) = 1 | + (−1.23 − 0.983i)2-s + (1.66 + 0.462i)3-s + (0.109 + 0.478i)4-s + (−0.0374 − 0.500i)5-s + (−1.60 − 2.21i)6-s + (−1.73 + 2.00i)7-s + (−1.03 + 2.14i)8-s + (2.57 + 1.54i)9-s + (−0.446 + 0.654i)10-s + (0.794 − 0.311i)11-s + (−0.0387 + 0.849i)12-s + (5.92 − 2.32i)13-s + (4.10 − 0.764i)14-s + (0.168 − 0.852i)15-s + (4.27 − 2.05i)16-s + (3.20 − 0.990i)17-s + ⋯ |
| L(s) = 1 | + (−0.872 − 0.695i)2-s + (0.963 + 0.266i)3-s + (0.0545 + 0.239i)4-s + (−0.0167 − 0.223i)5-s + (−0.655 − 0.903i)6-s + (−0.654 + 0.756i)7-s + (−0.365 + 0.758i)8-s + (0.857 + 0.514i)9-s + (−0.141 + 0.206i)10-s + (0.239 − 0.0940i)11-s + (−0.0111 + 0.245i)12-s + (1.64 − 0.644i)13-s + (1.09 − 0.204i)14-s + (0.0435 − 0.220i)15-s + (1.06 − 0.514i)16-s + (0.778 − 0.240i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16468 - 0.391891i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16468 - 0.391891i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.66 - 0.462i)T \) |
| 7 | \( 1 + (1.73 - 2.00i)T \) |
| good | 2 | \( 1 + (1.23 + 0.983i)T + (0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (0.0374 + 0.500i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-0.794 + 0.311i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-5.92 + 2.32i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-3.20 + 0.990i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-3.20 - 1.84i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.48 + 5.91i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (1.07 + 3.48i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 - 10.2iT - 31T^{2} \) |
| 37 | \( 1 + (-1.31 - 1.21i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-0.757 + 0.516i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (7.99 + 5.44i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (-0.242 + 0.303i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-6.63 - 7.15i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (7.75 - 3.73i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (4.12 + 0.942i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 - 4.42T + 67T^{2} \) |
| 71 | \( 1 + (8.81 - 2.01i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-9.85 - 3.86i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 - 3.72T + 79T^{2} \) |
| 83 | \( 1 + (-4.99 + 12.7i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-1.57 + 0.237i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (1.10 - 0.638i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.54410867075052501096530604532, −10.17423796788418442505543763063, −9.112410251864095749018710409334, −8.647449792648510507258738052951, −7.943800979495127265486938607253, −6.34658212961908625582716428312, −5.27654858714201299250295750949, −3.60213555597359894859119233488, −2.74295634789771723636658412169, −1.29841223367800849782578535826,
1.26185153332085723240690902349, 3.35531289435777988704617419130, 3.90824324310406123185014632055, 6.10619827324261622972777288103, 6.86275342218219812945076012219, 7.66172765451796143321876206311, 8.352618642873466517723870214839, 9.435824989902504249343932012542, 9.718011298063851995732220241850, 10.97967768492121575295102729028
