| L(s) = 1 | + (−0.505 − 1.32i)2-s + (−1.48 + 1.33i)4-s + (−3.21 − 1.16i)5-s + (−0.199 − 0.0351i)7-s + (2.51 + 1.28i)8-s + (0.0809 + 4.83i)10-s + (−1.37 − 3.78i)11-s + (1.09 + 1.30i)13-s + (0.0543 + 0.280i)14-s + (0.429 − 3.97i)16-s + (−1.57 + 0.908i)17-s + (−3.24 + 5.61i)19-s + (6.34 − 2.55i)20-s + (−4.29 + 3.73i)22-s + (1.37 + 7.80i)23-s + ⋯ |
| L(s) = 1 | + (−0.357 − 0.933i)2-s + (−0.744 + 0.668i)4-s + (−1.43 − 0.523i)5-s + (−0.0752 − 0.0132i)7-s + (0.890 + 0.455i)8-s + (0.0256 + 1.52i)10-s + (−0.415 − 1.14i)11-s + (0.304 + 0.362i)13-s + (0.0145 + 0.0750i)14-s + (0.107 − 0.994i)16-s + (−0.381 + 0.220i)17-s + (−0.743 + 1.28i)19-s + (1.41 − 0.570i)20-s + (−0.916 + 0.795i)22-s + (0.286 + 1.62i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.435330 + 0.138231i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.435330 + 0.138231i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.505 + 1.32i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.21 + 1.16i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.199 + 0.0351i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.37 + 3.78i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.09 - 1.30i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.57 - 0.908i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.24 - 5.61i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.37 - 7.80i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.08 - 2.58i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-7.51 + 1.32i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.53 + 0.885i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.61 - 7.88i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.47 + 1.26i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.523 - 2.97i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 9.46T + 53T^{2} \) |
| 59 | \( 1 + (1.23 - 3.38i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (8.13 + 1.43i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.26 - 1.06i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.24 - 3.89i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.05 - 7.01i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.380 - 0.453i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.02 + 9.56i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (10.7 + 6.18i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.24 - 3.00i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.94600366071810510926264676820, −9.827658094762895113923396570765, −8.811607861722397564091171900393, −8.168042456805968599085840842389, −7.64302043821191009796426825980, −6.09319685897862067927794454698, −4.71020274131655087958069124662, −3.88865240190925854707303789575, −3.04936145955810489589662464211, −1.22248674133032257733560301350,
0.32378575175869916216197526742, 2.71513865028857519626291853470, 4.32966761409140913607617194461, 4.73303328179046451238411711166, 6.38893857694040016005257662817, 6.96708687072265967426153735877, 7.82793037293793881134877035365, 8.441107169126671905370363644317, 9.433211646521550836043448348658, 10.55358074069951651817945073843
