L(s) = 1 | + (−0.104 + 0.994i)3-s + (−0.913 − 0.406i)4-s + (0.5 − 0.866i)5-s + (−0.978 − 0.207i)9-s + (0.978 + 0.207i)11-s + (0.499 − 0.866i)12-s + (0.809 + 0.587i)15-s + (0.669 + 0.743i)16-s + (−0.809 + 0.587i)20-s + (−0.873 − 0.786i)23-s + (−0.499 − 0.866i)25-s + (0.309 − 0.951i)27-s + (0.139 − 1.33i)31-s + (−0.309 + 0.951i)33-s + (0.809 + 0.587i)36-s + (1.64 + 0.535i)37-s + ⋯ |
L(s) = 1 | + (−0.104 + 0.994i)3-s + (−0.913 − 0.406i)4-s + (0.5 − 0.866i)5-s + (−0.978 − 0.207i)9-s + (0.978 + 0.207i)11-s + (0.499 − 0.866i)12-s + (0.809 + 0.587i)15-s + (0.669 + 0.743i)16-s + (−0.809 + 0.587i)20-s + (−0.873 − 0.786i)23-s + (−0.499 − 0.866i)25-s + (0.309 − 0.951i)27-s + (0.139 − 1.33i)31-s + (−0.309 + 0.951i)33-s + (0.809 + 0.587i)36-s + (1.64 + 0.535i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.001922585\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.001922585\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.104 - 0.994i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.978 - 0.207i)T \) |
good | 2 | \( 1 + (0.913 + 0.406i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.913 - 0.406i)T^{2} \) |
| 17 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.873 + 0.786i)T + (0.104 + 0.994i)T^{2} \) |
| 29 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 31 | \( 1 + (-0.139 + 1.33i)T + (-0.978 - 0.207i)T^{2} \) |
| 37 | \( 1 + (-1.64 - 0.535i)T + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-1.72 + 0.181i)T + (0.978 - 0.207i)T^{2} \) |
| 53 | \( 1 + (0.873 + 1.20i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-1.91 + 0.406i)T + (0.913 - 0.406i)T^{2} \) |
| 61 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 67 | \( 1 + (1.97 + 0.207i)T + (0.978 + 0.207i)T^{2} \) |
| 71 | \( 1 + (-1.58 + 1.14i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 83 | \( 1 + (0.669 - 0.743i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (1.47 - 0.155i)T + (0.978 - 0.207i)T^{2} \) |
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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
−9.282336835305981747057631084909, −8.571641436995710344086627827478, −7.920206265844331196227161606221, −6.32004849251717835809916928683, −5.88114692138764373357989021971, −4.99008339846206338410183181640, −4.31539472862584572286356981873, −3.84996962426151276846610310981, −2.29670674998840505579042732880, −0.834992779307046965694878795954,
1.21244203566417837792865106067, 2.44760806639702394957432294059, 3.38055460337315620688916891458, 4.24276179202808215630019357230, 5.54523806285105024616385527347, 6.02476947371123446579212162410, 6.98326655803548513397103937933, 7.50341212375953583046582336304, 8.384695166534064123183478630893, 9.069131148863050853390382671875