| L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 7-s − 8-s + 9-s + 3·10-s + 2·11-s + 12-s + 13-s + 14-s − 3·15-s + 16-s − 17-s − 18-s + 4·19-s − 3·20-s − 21-s − 2·22-s − 6·23-s − 24-s + 4·25-s − 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.603·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.670·20-s − 0.218·21-s − 0.426·22-s − 1.25·23-s − 0.204·24-s + 4/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 911 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.36220635000808, −14.04721172438906, −13.36080417087322, −12.79009825458032, −12.17587153659569, −11.88673837626445, −11.38129608451722, −10.83813633450565, −10.32436994366414, −9.684507254509232, −9.194065882807014, −8.811975968065096, −8.220846964613647, −7.730039132540802, −7.362820863648986, −6.827434003501912, −6.265596736505758, −5.495968991022327, −4.836846704914696, −3.943594219275252, −3.530874727829711, −3.364590297761834, −2.196809332526509, −1.736160165732874, −0.7473403531978404, 0,
0.7473403531978404, 1.736160165732874, 2.196809332526509, 3.364590297761834, 3.530874727829711, 3.943594219275252, 4.836846704914696, 5.495968991022327, 6.265596736505758, 6.827434003501912, 7.362820863648986, 7.730039132540802, 8.220846964613647, 8.811975968065096, 9.194065882807014, 9.684507254509232, 10.32436994366414, 10.83813633450565, 11.38129608451722, 11.88673837626445, 12.17587153659569, 12.79009825458032, 13.36080417087322, 14.04721172438906, 14.36220635000808
