| L(s) = 1 | + 1.34·2-s − 3-s − 0.196·4-s + 1.69·5-s − 1.34·6-s + 7-s − 2.94·8-s + 9-s + 2.27·10-s − 0.941·11-s + 0.196·12-s − 0.611·13-s + 1.34·14-s − 1.69·15-s − 3.56·16-s + 4.60·17-s + 1.34·18-s − 1.78·19-s − 0.332·20-s − 21-s − 1.26·22-s + 0.502·23-s + 2.94·24-s − 2.13·25-s − 0.821·26-s − 27-s − 0.196·28-s + ⋯ |
| L(s) = 1 | + 0.949·2-s − 0.577·3-s − 0.0981·4-s + 0.756·5-s − 0.548·6-s + 0.377·7-s − 1.04·8-s + 0.333·9-s + 0.718·10-s − 0.283·11-s + 0.0566·12-s − 0.169·13-s + 0.358·14-s − 0.436·15-s − 0.892·16-s + 1.11·17-s + 0.316·18-s − 0.410·19-s − 0.0742·20-s − 0.218·21-s − 0.269·22-s + 0.104·23-s + 0.602·24-s − 0.427·25-s − 0.161·26-s − 0.192·27-s − 0.0370·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 5 | \( 1 - 1.69T + 5T^{2} \) |
| 11 | \( 1 + 0.941T + 11T^{2} \) |
| 13 | \( 1 + 0.611T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 + 1.78T + 19T^{2} \) |
| 23 | \( 1 - 0.502T + 23T^{2} \) |
| 29 | \( 1 + 5.64T + 29T^{2} \) |
| 31 | \( 1 + 4.32T + 31T^{2} \) |
| 37 | \( 1 - 9.15T + 37T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 + 3.50T + 43T^{2} \) |
| 47 | \( 1 - 0.559T + 47T^{2} \) |
| 53 | \( 1 + 4.66T + 53T^{2} \) |
| 59 | \( 1 + 2.59T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 5.02T + 67T^{2} \) |
| 71 | \( 1 + 8.96T + 71T^{2} \) |
| 73 | \( 1 + 8.37T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 1.76T + 83T^{2} \) |
| 89 | \( 1 + 4.67T + 89T^{2} \) |
| 97 | \( 1 + 4.52T + 97T^{2} \) |
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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
−7.45829140663537587750778518926, −6.39753202840673571660801091991, −5.88346476394278247844755400531, −5.42376432328549663280976424603, −4.78582215698566541027740790655, −4.07445098846931137935700666973, −3.26418027163091769534310690772, −2.34140339261081905542626467431, −1.36402116604148406148976842994, 0,
1.36402116604148406148976842994, 2.34140339261081905542626467431, 3.26418027163091769534310690772, 4.07445098846931137935700666973, 4.78582215698566541027740790655, 5.42376432328549663280976424603, 5.88346476394278247844755400531, 6.39753202840673571660801091991, 7.45829140663537587750778518926
