L(s) = 1 | + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (−0.5 − 0.866i)5-s + (0.669 − 0.743i)9-s + (0.669 − 0.743i)11-s + (−0.499 − 0.866i)12-s + (−0.809 − 0.587i)15-s + (−0.978 + 0.207i)16-s + (−0.809 + 0.587i)20-s + (1.58 + 0.336i)23-s + (−0.499 + 0.866i)25-s + (0.309 − 0.951i)27-s + (−1.78 + 0.795i)31-s + (0.309 − 0.951i)33-s + (−0.809 − 0.587i)36-s + (−0.309 + 0.951i)37-s + ⋯ |
L(s) = 1 | + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (−0.5 − 0.866i)5-s + (0.669 − 0.743i)9-s + (0.669 − 0.743i)11-s + (−0.499 − 0.866i)12-s + (−0.809 − 0.587i)15-s + (−0.978 + 0.207i)16-s + (−0.809 + 0.587i)20-s + (1.58 + 0.336i)23-s + (−0.499 + 0.866i)25-s + (0.309 − 0.951i)27-s + (−1.78 + 0.795i)31-s + (0.309 − 0.951i)33-s + (−0.809 − 0.587i)36-s + (−0.309 + 0.951i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.500787408\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.500787408\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.913 + 0.406i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.669 + 0.743i)T \) |
good | 2 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-1.58 - 0.336i)T + (0.913 + 0.406i)T^{2} \) |
| 29 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 31 | \( 1 + (1.78 - 0.795i)T + (0.669 - 0.743i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.913 + 0.406i)T + (0.669 + 0.743i)T^{2} \) |
| 53 | \( 1 + (-1.58 + 1.14i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.895 - 0.994i)T + (-0.104 + 0.994i)T^{2} \) |
| 61 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 67 | \( 1 + (-1.66 + 0.743i)T + (0.669 - 0.743i)T^{2} \) |
| 71 | \( 1 + (1.08 - 0.786i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 83 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 89 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (1.78 + 0.795i)T + (0.669 + 0.743i)T^{2} \) |
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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
−8.795688757975369419475460900245, −8.427743379078500848758180047916, −7.26091325203933587151235270435, −6.74253358882940883334243726954, −5.65430513159977301787594127303, −4.94872911403720636872314112518, −3.96089362253465727851993115003, −3.16029873927837858612388533760, −1.74005489191758040511059021569, −0.968592042112737796904754500208,
2.06634662152155379846176729408, 2.89974706744363516036736577533, 3.74161533658578293843450524741, 4.17180796227022317162909903383, 5.22825898077336767136890959731, 6.79014803186273536191859444710, 7.15148281886275975386310181232, 7.81218639123244602422467905393, 8.619158226045608849501483923521, 9.233626523991961905630699585563