L(s) = 1 | + (−1.08 + 0.786i)2-s + (0.913 − 0.406i)3-s + (0.244 − 0.752i)4-s + (−0.5 − 0.866i)5-s + (−0.669 + 1.15i)6-s + (−0.0864 − 0.266i)8-s + (0.669 − 0.743i)9-s + (1.22 + 0.544i)10-s + (−0.0826 − 0.786i)12-s + (−0.809 − 0.587i)15-s + (0.942 + 0.684i)16-s + (0.978 + 0.207i)17-s + (−0.139 + 1.33i)18-s + (−0.190 − 1.81i)19-s + (−0.773 + 0.164i)20-s + ⋯ |
L(s) = 1 | + (−1.08 + 0.786i)2-s + (0.913 − 0.406i)3-s + (0.244 − 0.752i)4-s + (−0.5 − 0.866i)5-s + (−0.669 + 1.15i)6-s + (−0.0864 − 0.266i)8-s + (0.669 − 0.743i)9-s + (1.22 + 0.544i)10-s + (−0.0826 − 0.786i)12-s + (−0.809 − 0.587i)15-s + (0.942 + 0.684i)16-s + (0.978 + 0.207i)17-s + (−0.139 + 1.33i)18-s + (−0.190 − 1.81i)19-s + (−0.773 + 0.164i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6253626705\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6253626705\) |
\(L(1)\) |
|
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 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
good | 2 | \( 1 + (1.08 - 0.786i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 11 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 13 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 17 | \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \) |
| 19 | \( 1 + (0.190 + 1.81i)T + (-0.978 + 0.207i)T^{2} \) |
| 23 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 43 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 47 | \( 1 + (-1.58 - 1.14i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (1.08 - 1.20i)T + (-0.104 - 0.994i)T^{2} \) |
| 59 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 61 | \( 1 + 0.209T + T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 73 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 79 | \( 1 + (1.78 + 0.379i)T + (0.913 + 0.406i)T^{2} \) |
| 83 | \( 1 + (1.78 + 0.795i)T + (0.669 + 0.743i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.809 + 0.587i)T^{2} \) |
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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
−11.07283103960470411595092272229, −9.793322053313153488813054589716, −9.045627248175468177848311491793, −8.636472523238773515303431972511, −7.56734088817262365215847858468, −7.26232051637319377363047794551, −5.92302420414438655466067181038, −4.44586980775492035884104680437, −3.16599391802978997064997536895, −1.21024690129198319322793724669,
1.91777998220283602764779490013, 3.06962902934039014613600106614, 3.94066321158446791085238029042, 5.61679569463955218616839897012, 7.20085316986539824033347895181, 8.005885191728837618676685058741, 8.609677414663336973154452671432, 9.791305655349875781490106217491, 10.16857628694377829797713106269, 10.94100619444204532770998084040