| L(s) = 1 | + (0.104 − 0.994i)2-s + (0.669 + 0.743i)3-s + (−0.978 − 0.207i)4-s + (−3.04 − 1.35i)5-s + (0.809 − 0.587i)6-s + (0.194 + 2.63i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (−1.66 + 2.88i)10-s + (−0.465 + 3.28i)11-s + (−0.5 − 0.866i)12-s + (4.99 + 3.62i)13-s + (2.64 + 0.0818i)14-s + (−1.02 − 3.16i)15-s + (0.913 + 0.406i)16-s + (0.288 + 2.74i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 − 0.703i)2-s + (0.386 + 0.429i)3-s + (−0.489 − 0.103i)4-s + (−1.36 − 0.605i)5-s + (0.330 − 0.239i)6-s + (0.0737 + 0.997i)7-s + (−0.109 + 0.336i)8-s + (−0.0348 + 0.331i)9-s + (−0.526 + 0.911i)10-s + (−0.140 + 0.990i)11-s + (−0.144 − 0.249i)12-s + (1.38 + 1.00i)13-s + (0.706 + 0.0218i)14-s + (−0.265 − 0.817i)15-s + (0.228 + 0.101i)16-s + (0.0699 + 0.665i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00004 + 0.444142i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00004 + 0.444142i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 + 0.994i)T \) |
| 3 | \( 1 + (-0.669 - 0.743i)T \) |
| 7 | \( 1 + (-0.194 - 2.63i)T \) |
| 11 | \( 1 + (0.465 - 3.28i)T \) |
| good | 5 | \( 1 + (3.04 + 1.35i)T + (3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (-4.99 - 3.62i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.288 - 2.74i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-1.87 + 0.397i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (1.57 + 2.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.159 + 0.491i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.36 + 0.606i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (1.86 - 2.07i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (3.61 - 11.1i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + (2.35 - 0.500i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-0.815 + 0.363i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (1.54 + 0.327i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-13.6 - 6.08i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (0.0892 - 0.154i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.96 + 6.51i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (3.59 + 0.763i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-0.932 + 8.86i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-8.79 + 6.38i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.291 - 0.504i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.88 - 7.17i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.53481961354821078137713597081, −10.29510183421556156068507692717, −9.311030799230755805301946226011, −8.509636677241764767823086794859, −8.031042559955123821781341216513, −6.48494948877689948626101357486, −4.99323618679506000551320872759, −4.23939989277643472072171363556, −3.32160382962057613745396587923, −1.76240892233770170500621451230,
0.67534657231974088960706502042, 3.43017304569010860718489629986, 3.67523288322405724131837953865, 5.33125628871118135563091434314, 6.57510262049260597297891508591, 7.35249565529932822345283912819, 8.062695808142072526527734693984, 8.604275544975070734843388275987, 10.11076743918487809751413534100, 11.04340160991925450364549376862
