| L(s) = 1 | − 2.12·2-s + 3-s + 2.53·4-s − 4.00·5-s − 2.12·6-s + 2.85·7-s − 1.13·8-s + 9-s + 8.53·10-s + 0.813·11-s + 2.53·12-s − 0.802·13-s − 6.07·14-s − 4.00·15-s − 2.65·16-s − 17-s − 2.12·18-s + 1.48·19-s − 10.1·20-s + 2.85·21-s − 1.73·22-s + 0.00429·23-s − 1.13·24-s + 11.0·25-s + 1.70·26-s + 27-s + 7.22·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s + 0.577·3-s + 1.26·4-s − 1.79·5-s − 0.869·6-s + 1.07·7-s − 0.400·8-s + 0.333·9-s + 2.69·10-s + 0.245·11-s + 0.731·12-s − 0.222·13-s − 1.62·14-s − 1.03·15-s − 0.663·16-s − 0.242·17-s − 0.501·18-s + 0.340·19-s − 2.26·20-s + 0.622·21-s − 0.369·22-s + 0.000894·23-s − 0.231·24-s + 2.21·25-s + 0.335·26-s + 0.192·27-s + 1.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 5 | \( 1 + 4.00T + 5T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 11 | \( 1 - 0.813T + 11T^{2} \) |
| 13 | \( 1 + 0.802T + 13T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 23 | \( 1 - 0.00429T + 23T^{2} \) |
| 29 | \( 1 + 4.50T + 29T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 2.58T + 43T^{2} \) |
| 47 | \( 1 + 3.09T + 47T^{2} \) |
| 53 | \( 1 - 0.598T + 53T^{2} \) |
| 59 | \( 1 - 4.12T + 59T^{2} \) |
| 61 | \( 1 - 5.03T + 61T^{2} \) |
| 67 | \( 1 + 1.96T + 67T^{2} \) |
| 71 | \( 1 - 8.01T + 71T^{2} \) |
| 73 | \( 1 - 3.22T + 73T^{2} \) |
| 79 | \( 1 - 1.07T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 5.84T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−7.84106887957009022427762579801, −7.13558530336010438096397428595, −6.74241112711348495874911351388, −5.23629216616240671215172074552, −4.49009946428348511632338501583, −3.88306130373375743911508184559, −2.96747687154175068558333133042, −1.92095794799617365747409828595, −1.06358501619639505313832618390, 0,
1.06358501619639505313832618390, 1.92095794799617365747409828595, 2.96747687154175068558333133042, 3.88306130373375743911508184559, 4.49009946428348511632338501583, 5.23629216616240671215172074552, 6.74241112711348495874911351388, 7.13558530336010438096397428595, 7.84106887957009022427762579801
