| L(s) = 1 | + 3-s + 0.413·5-s + 2.72·7-s + 9-s − 3.29·11-s − 3.51·13-s + 0.413·15-s − 2.04·17-s + 0.104·19-s + 2.72·21-s − 7.14·23-s − 4.82·25-s + 27-s + 3.94·29-s − 6.72·31-s − 3.29·33-s + 1.12·35-s − 11.8·37-s − 3.51·39-s + 1.68·41-s − 3.25·43-s + 0.413·45-s − 1.28·47-s + 0.427·49-s − 2.04·51-s − 2.39·53-s − 1.36·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.185·5-s + 1.03·7-s + 0.333·9-s − 0.992·11-s − 0.974·13-s + 0.106·15-s − 0.495·17-s + 0.0239·19-s + 0.594·21-s − 1.49·23-s − 0.965·25-s + 0.192·27-s + 0.733·29-s − 1.20·31-s − 0.573·33-s + 0.190·35-s − 1.94·37-s − 0.562·39-s + 0.263·41-s − 0.495·43-s + 0.0616·45-s − 0.187·47-s + 0.0610·49-s − 0.286·51-s − 0.328·53-s − 0.183·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 0.413T + 5T^{2} \) |
| 7 | \( 1 - 2.72T + 7T^{2} \) |
| 11 | \( 1 + 3.29T + 11T^{2} \) |
| 13 | \( 1 + 3.51T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 - 0.104T + 19T^{2} \) |
| 23 | \( 1 + 7.14T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 + 6.72T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 + 3.25T + 43T^{2} \) |
| 47 | \( 1 + 1.28T + 47T^{2} \) |
| 53 | \( 1 + 2.39T + 53T^{2} \) |
| 59 | \( 1 - 4.83T + 59T^{2} \) |
| 61 | \( 1 + 0.869T + 61T^{2} \) |
| 67 | \( 1 - 0.884T + 67T^{2} \) |
| 71 | \( 1 - 5.76T + 71T^{2} \) |
| 73 | \( 1 + 9.55T + 73T^{2} \) |
| 79 | \( 1 - 8.30T + 79T^{2} \) |
| 83 | \( 1 + 9.73T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
| 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
−8.047712377696303594423930534243, −7.56727177453783531374626081508, −6.78507900216073861859975559366, −5.68502139194502149534383658410, −5.04918407102090649505057626287, −4.33331891110010956105727980367, −3.36650408746486904357079300050, −2.24303317049076766486315318735, −1.81871984208376143198129717124, 0,
1.81871984208376143198129717124, 2.24303317049076766486315318735, 3.36650408746486904357079300050, 4.33331891110010956105727980367, 5.04918407102090649505057626287, 5.68502139194502149534383658410, 6.78507900216073861859975559366, 7.56727177453783531374626081508, 8.047712377696303594423930534243
