L(s) = 1 | + 1.84·2-s + 1.50·3-s + 1.40·4-s + 5-s + 2.78·6-s + 7-s − 1.10·8-s − 0.725·9-s + 1.84·10-s − 0.364·11-s + 2.11·12-s − 4.79·13-s + 1.84·14-s + 1.50·15-s − 4.83·16-s − 2.87·17-s − 1.33·18-s − 4.60·19-s + 1.40·20-s + 1.50·21-s − 0.671·22-s + 0.362·23-s − 1.66·24-s + 25-s − 8.85·26-s − 5.61·27-s + 1.40·28-s + ⋯ |
L(s) = 1 | + 1.30·2-s + 0.870·3-s + 0.700·4-s + 0.447·5-s + 1.13·6-s + 0.377·7-s − 0.390·8-s − 0.241·9-s + 0.583·10-s − 0.109·11-s + 0.610·12-s − 1.33·13-s + 0.492·14-s + 0.389·15-s − 1.20·16-s − 0.696·17-s − 0.315·18-s − 1.05·19-s + 0.313·20-s + 0.329·21-s − 0.143·22-s + 0.0755·23-s − 0.339·24-s + 0.200·25-s − 1.73·26-s − 1.08·27-s + 0.264·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 - 1.84T + 2T^{2} \) |
| 3 | \( 1 - 1.50T + 3T^{2} \) |
| 11 | \( 1 + 0.364T + 11T^{2} \) |
| 13 | \( 1 + 4.79T + 13T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 + 4.60T + 19T^{2} \) |
| 23 | \( 1 - 0.362T + 23T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 - 6.71T + 31T^{2} \) |
| 37 | \( 1 - 7.09T + 37T^{2} \) |
| 41 | \( 1 + 0.0307T + 41T^{2} \) |
| 43 | \( 1 + 6.19T + 43T^{2} \) |
| 47 | \( 1 - 5.81T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 + 4.40T + 59T^{2} \) |
| 61 | \( 1 - 2.83T + 61T^{2} \) |
| 67 | \( 1 + 4.31T + 67T^{2} \) |
| 71 | \( 1 + 4.80T + 71T^{2} \) |
| 73 | \( 1 - 5.26T + 73T^{2} \) |
| 79 | \( 1 + 3.24T + 79T^{2} \) |
| 83 | \( 1 + 8.11T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 9.86T + 97T^{2} \) |
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
−7.41598060453409341447758231383, −6.58614622768600461480138971539, −6.01036003268151823967465501144, −5.21614999989209080678484702120, −4.57187943647163885512521857750, −4.08246880080575801728316503713, −2.93712652417837388125750066131, −2.62021939524974728738034808384, −1.84556677137016756924834903139, 0,
1.84556677137016756924834903139, 2.62021939524974728738034808384, 2.93712652417837388125750066131, 4.08246880080575801728316503713, 4.57187943647163885512521857750, 5.21614999989209080678484702120, 6.01036003268151823967465501144, 6.58614622768600461480138971539, 7.41598060453409341447758231383