L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.110 − 1.74i)3-s + (0.309 + 0.951i)4-s + (−1.11 + 1.35i)6-s + (0.309 − 0.951i)8-s + (−2.05 − 0.259i)9-s + (−1.80 + 0.462i)11-s + (1.69 − 0.435i)12-s + (−0.809 + 0.587i)16-s + (−1.27 − 1.54i)17-s + (1.51 + 1.41i)18-s + (0.688 + 1.46i)19-s + (1.72 + 0.684i)22-s + (−1.62 − 0.645i)24-s + (−0.809 − 0.587i)25-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.110 − 1.74i)3-s + (0.309 + 0.951i)4-s + (−1.11 + 1.35i)6-s + (0.309 − 0.951i)8-s + (−2.05 − 0.259i)9-s + (−1.80 + 0.462i)11-s + (1.69 − 0.435i)12-s + (−0.809 + 0.587i)16-s + (−1.27 − 1.54i)17-s + (1.51 + 1.41i)18-s + (0.688 + 1.46i)19-s + (1.72 + 0.684i)22-s + (−1.62 − 0.645i)24-s + (−0.809 − 0.587i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2046185429\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2046185429\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 251 | \( 1 + (0.187 + 0.982i)T \) |
good | 3 | \( 1 + (-0.110 + 1.74i)T + (-0.992 - 0.125i)T^{2} \) |
| 5 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (0.929 + 0.368i)T^{2} \) |
| 11 | \( 1 + (1.80 - 0.462i)T + (0.876 - 0.481i)T^{2} \) |
| 13 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 17 | \( 1 + (1.27 + 1.54i)T + (-0.187 + 0.982i)T^{2} \) |
| 19 | \( 1 + (-0.688 - 1.46i)T + (-0.637 + 0.770i)T^{2} \) |
| 23 | \( 1 + (0.187 - 0.982i)T^{2} \) |
| 29 | \( 1 + (-0.0627 + 0.998i)T^{2} \) |
| 31 | \( 1 + (-0.876 + 0.481i)T^{2} \) |
| 37 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 41 | \( 1 + (-0.348 + 0.137i)T + (0.728 - 0.684i)T^{2} \) |
| 43 | \( 1 + (1.11 - 0.614i)T + (0.535 - 0.844i)T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.535 - 0.844i)T^{2} \) |
| 59 | \( 1 + (1.23 - 1.49i)T + (-0.187 - 0.982i)T^{2} \) |
| 61 | \( 1 + (-0.535 - 0.844i)T^{2} \) |
| 67 | \( 1 + (-0.791 + 0.313i)T + (0.728 - 0.684i)T^{2} \) |
| 71 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 73 | \( 1 + (0.101 + 1.61i)T + (-0.992 + 0.125i)T^{2} \) |
| 79 | \( 1 + (-0.0627 - 0.998i)T^{2} \) |
| 83 | \( 1 + (-0.0672 + 0.106i)T + (-0.425 - 0.904i)T^{2} \) |
| 89 | \( 1 + (0.929 + 1.12i)T + (-0.187 + 0.982i)T^{2} \) |
| 97 | \( 1 + (0.200 - 0.316i)T + (-0.425 - 0.904i)T^{2} \) |
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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
−8.546556419878836789365592863452, −7.84524590679077389087992719967, −7.53623843397294162072406804479, −6.77890873794029807050874748086, −5.86227545459758213070292920977, −4.72571961844385336081784809368, −3.13119229863293148914572631714, −2.42952013723160627740892185845, −1.67508834049778948185072242269, −0.17238931423082641288108348808,
2.27821206917197101219200818097, 3.27718304299454088954917741462, 4.47644366548105229594995698108, 5.16667284824824943485588713440, 5.75956975131110932352983886108, 6.79193776001004992781966678755, 7.976874558029921450050238403053, 8.418801281058558803669138884466, 9.203717279065887827727199510999, 9.795990438850241049561269700640