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
−11.04340160991925450364549376862, −10.11076743918487809751413534100, −8.604275544975070734843388275987, −8.062695808142072526527734693984, −7.35249565529932822345283912819, −6.57510262049260597297891508591, −5.33125628871118135563091434314, −3.67523288322405724131837953865, −3.43017304569010860718489629986, −0.67534657231974088960706502042,
1.76240892233770170500621451230, 3.32160382962057613745396587923, 4.23939989277643472072171363556, 4.99323618679506000551320872759, 6.48494948877689948626101357486, 8.031042559955123821781341216513, 8.509636677241764767823086794859, 9.311030799230755805301946226011, 10.29510183421556156068507692717, 11.53481961354821078137713597081