| L(s) = 1 | + 0.444·2-s − 3-s − 1.80·4-s + 0.495·5-s − 0.444·6-s + 2.63·7-s − 1.68·8-s + 9-s + 0.220·10-s + 4.65·11-s + 1.80·12-s − 13-s + 1.17·14-s − 0.495·15-s + 2.85·16-s − 8.19·17-s + 0.444·18-s − 0.642·19-s − 0.893·20-s − 2.63·21-s + 2.06·22-s − 4.89·23-s + 1.68·24-s − 4.75·25-s − 0.444·26-s − 27-s − 4.75·28-s + ⋯ |
| L(s) = 1 | + 0.314·2-s − 0.577·3-s − 0.901·4-s + 0.221·5-s − 0.181·6-s + 0.996·7-s − 0.597·8-s + 0.333·9-s + 0.0695·10-s + 1.40·11-s + 0.520·12-s − 0.277·13-s + 0.313·14-s − 0.127·15-s + 0.713·16-s − 1.98·17-s + 0.104·18-s − 0.147·19-s − 0.199·20-s − 0.575·21-s + 0.440·22-s − 1.02·23-s + 0.344·24-s − 0.950·25-s − 0.0871·26-s − 0.192·27-s − 0.898·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 0.444T + 2T^{2} \) |
| 5 | \( 1 - 0.495T + 5T^{2} \) |
| 7 | \( 1 - 2.63T + 7T^{2} \) |
| 11 | \( 1 - 4.65T + 11T^{2} \) |
| 17 | \( 1 + 8.19T + 17T^{2} \) |
| 19 | \( 1 + 0.642T + 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 + 6.98T + 29T^{2} \) |
| 31 | \( 1 - 5.25T + 31T^{2} \) |
| 37 | \( 1 - 6.45T + 37T^{2} \) |
| 41 | \( 1 - 0.00128T + 41T^{2} \) |
| 43 | \( 1 - 3.72T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 7.76T + 53T^{2} \) |
| 59 | \( 1 + 4.72T + 59T^{2} \) |
| 61 | \( 1 + 1.19T + 61T^{2} \) |
| 67 | \( 1 + 2.43T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 17.8T + 83T^{2} \) |
| 89 | \( 1 + 3.73T + 89T^{2} \) |
| 97 | \( 1 + 2.19T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.088184414374010205342938292270, −7.37445066838075310216119940510, −6.23093138825446864522258533750, −5.98986764053106397410703071955, −4.88426215130295224255234592640, −4.31278696698112262237281072866, −3.90908447779326650819455560917, −2.33044605503309677329785101797, −1.37677241807824802563864101157, 0,
1.37677241807824802563864101157, 2.33044605503309677329785101797, 3.90908447779326650819455560917, 4.31278696698112262237281072866, 4.88426215130295224255234592640, 5.98986764053106397410703071955, 6.23093138825446864522258533750, 7.37445066838075310216119940510, 8.088184414374010205342938292270
