L(s) = 1 | + (0.786 − 0.618i)2-s + (0.308 − 0.356i)3-s + (0.235 − 0.971i)4-s + (0.841 − 0.540i)5-s + (0.0224 − 0.470i)6-s + (−1.82 − 0.729i)7-s + (−0.415 − 0.909i)8-s + (0.110 + 0.769i)9-s + (0.327 − 0.945i)10-s + (−0.273 − 0.384i)12-s + (−1.88 + 0.553i)14-s + (0.0671 − 0.466i)15-s + (−0.888 − 0.458i)16-s + (0.562 + 0.536i)18-s + (−0.327 − 0.945i)20-s + (−0.823 + 0.424i)21-s + ⋯ |
L(s) = 1 | + (0.786 − 0.618i)2-s + (0.308 − 0.356i)3-s + (0.235 − 0.971i)4-s + (0.841 − 0.540i)5-s + (0.0224 − 0.470i)6-s + (−1.82 − 0.729i)7-s + (−0.415 − 0.909i)8-s + (0.110 + 0.769i)9-s + (0.327 − 0.945i)10-s + (−0.273 − 0.384i)12-s + (−1.88 + 0.553i)14-s + (0.0671 − 0.466i)15-s + (−0.888 − 0.458i)16-s + (0.562 + 0.536i)18-s + (−0.327 − 0.945i)20-s + (−0.823 + 0.424i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.721617517\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.721617517\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.786 + 0.618i)T \) |
| 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 67 | \( 1 + (0.723 + 0.690i)T \) |
good | 3 | \( 1 + (-0.308 + 0.356i)T + (-0.142 - 0.989i)T^{2} \) |
| 7 | \( 1 + (1.82 + 0.729i)T + (0.723 + 0.690i)T^{2} \) |
| 11 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 13 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 17 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 19 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 23 | \( 1 + (-0.981 - 0.189i)T + (0.928 + 0.371i)T^{2} \) |
| 29 | \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (1.44 - 1.37i)T + (0.0475 - 0.998i)T^{2} \) |
| 43 | \( 1 + (-1.38 - 0.407i)T + (0.841 + 0.540i)T^{2} \) |
| 47 | \( 1 + (-0.379 - 1.09i)T + (-0.786 + 0.618i)T^{2} \) |
| 53 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 59 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 61 | \( 1 + (-0.0800 + 1.68i)T + (-0.995 - 0.0950i)T^{2} \) |
| 71 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 73 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 79 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 83 | \( 1 + (-1.65 - 0.850i)T + (0.580 + 0.814i)T^{2} \) |
| 89 | \( 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.601071538903982637470201896838, −9.181256992221836615626112644557, −7.82369760466297257031077152487, −6.73244335533118272853189712483, −6.29111464089750306735130703626, −5.25733319459789631786428973695, −4.37340711164039077231170421225, −3.23165459207035663154071717103, −2.48844073571243632720252706311, −1.16684624598328485871010625832,
2.50880051096069469309482483205, 3.15896678130244818813328277504, 3.88029195569032923879946130121, 5.31692440640738039651620044292, 5.97010639903671870072200345593, 6.73470950955782335664576628114, 7.11668680820760749883278831081, 8.881754926058114827281190016861, 8.979466344209587063419084188197, 9.967232811442064801791174958283