| L(s) = 1 | + (1.46 − 0.220i)2-s + (1.61 + 0.630i)3-s + (0.174 − 0.0538i)4-s + (−0.764 − 0.520i)5-s + (2.49 + 0.565i)6-s + (1.69 + 2.02i)7-s + (−2.41 + 1.16i)8-s + (2.20 + 2.03i)9-s + (−1.23 − 0.592i)10-s + (2.99 − 0.452i)11-s + (0.315 + 0.0231i)12-s + (1.91 − 4.87i)13-s + (2.92 + 2.58i)14-s + (−0.904 − 1.32i)15-s + (−3.57 + 2.44i)16-s + (0.918 + 4.02i)17-s + ⋯ |
| L(s) = 1 | + (1.03 − 0.155i)2-s + (0.931 + 0.364i)3-s + (0.0872 − 0.0269i)4-s + (−0.341 − 0.232i)5-s + (1.01 + 0.231i)6-s + (0.642 + 0.766i)7-s + (−0.855 + 0.411i)8-s + (0.734 + 0.678i)9-s + (−0.389 − 0.187i)10-s + (0.904 − 0.136i)11-s + (0.0910 + 0.00669i)12-s + (0.530 − 1.35i)13-s + (0.782 + 0.691i)14-s + (−0.233 − 0.341i)15-s + (−0.894 + 0.610i)16-s + (0.222 + 0.976i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.80157 + 0.536893i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.80157 + 0.536893i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.61 - 0.630i)T \) |
| 7 | \( 1 + (-1.69 - 2.02i)T \) |
| good | 2 | \( 1 + (-1.46 + 0.220i)T + (1.91 - 0.589i)T^{2} \) |
| 5 | \( 1 + (0.764 + 0.520i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-2.99 + 0.452i)T + (10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (-1.91 + 4.87i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-0.918 - 4.02i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 - 0.0896T + 19T^{2} \) |
| 23 | \( 1 + (3.67 - 1.13i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (1.76 + 0.544i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (3.48 + 6.04i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.83 + 8.01i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (7.22 + 4.92i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (9.44 - 6.44i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (-3.21 + 0.484i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-0.412 + 1.80i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (-0.684 - 9.13i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-11.2 - 3.47i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (5.46 + 9.45i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.631 + 2.76i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-6.80 + 8.53i)T + (-16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 + (1.75 - 3.04i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.37 - 11.1i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (0.598 - 0.750i)T + (-19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 + (-7.64 + 13.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.42796863891638889295085906371, −10.32494772638003536111226222417, −9.159399898367652415926786575448, −8.438975458650229810902354231582, −7.85548719347444103430472145425, −6.07519040336283162438579605040, −5.24901912563826090178642682768, −4.06395056881569236852754108167, −3.48898978851842121996529153929, −2.08832528245491545912502034249,
1.61105257541142857950961050557, 3.42329285056652933008403536006, 4.01872049141278664315562380935, 5.01050401405225169941078602617, 6.66830092221722856341875658336, 7.02871002112326084385886676769, 8.325373400810553132924836576788, 9.152200071998304099156857052236, 10.02513257078234421784534548674, 11.58915404408833081668505362900
