| L(s) = 1 | − 3-s + 3.23·5-s − 7-s + 9-s + 3.68·11-s − 0.277·13-s − 3.23·15-s + 2.17·17-s + 19-s + 21-s − 2.27·23-s + 5.47·25-s − 27-s − 1.51·29-s + 1.40·31-s − 3.68·33-s − 3.23·35-s + 7.96·37-s + 0.277·39-s + 9.02·41-s − 4.55·43-s + 3.23·45-s − 2.44·47-s + 49-s − 2.17·51-s + 5.51·53-s + 11.9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.44·5-s − 0.377·7-s + 0.333·9-s + 1.11·11-s − 0.0768·13-s − 0.835·15-s + 0.526·17-s + 0.229·19-s + 0.218·21-s − 0.474·23-s + 1.09·25-s − 0.192·27-s − 0.280·29-s + 0.252·31-s − 0.641·33-s − 0.546·35-s + 1.30·37-s + 0.0443·39-s + 1.40·41-s − 0.694·43-s + 0.482·45-s − 0.357·47-s + 0.142·49-s − 0.304·51-s + 0.757·53-s + 1.60·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.414458794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.414458794\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 - 3.68T + 11T^{2} \) |
| 13 | \( 1 + 0.277T + 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 23 | \( 1 + 2.27T + 23T^{2} \) |
| 29 | \( 1 + 1.51T + 29T^{2} \) |
| 31 | \( 1 - 1.40T + 31T^{2} \) |
| 37 | \( 1 - 7.96T + 37T^{2} \) |
| 41 | \( 1 - 9.02T + 41T^{2} \) |
| 43 | \( 1 + 4.55T + 43T^{2} \) |
| 47 | \( 1 + 2.44T + 47T^{2} \) |
| 53 | \( 1 - 5.51T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 2.55T + 61T^{2} \) |
| 67 | \( 1 - 3.68T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 2.34T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 1.38T + 83T^{2} \) |
| 89 | \( 1 - 2.34T + 89T^{2} \) |
| 97 | \( 1 - 9.73T + 97T^{2} \) |
| 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
−7.966548550907297169659870923676, −7.12983767246536387709142900956, −6.34906357288551867189598599144, −6.03601790297032983489986456364, −5.37634371275599189726219002560, −4.50812660604702416756332265142, −3.65865484050735082629756586696, −2.62829728583878818809931398311, −1.73026528070027403657501969967, −0.877910889428461346092927723888,
0.877910889428461346092927723888, 1.73026528070027403657501969967, 2.62829728583878818809931398311, 3.65865484050735082629756586696, 4.50812660604702416756332265142, 5.37634371275599189726219002560, 6.03601790297032983489986456364, 6.34906357288551867189598599144, 7.12983767246536387709142900956, 7.966548550907297169659870923676
