| L(s) = 1 | − 2-s + 3-s + 4-s + 2.36·5-s − 6-s − 4.26·7-s − 8-s + 9-s − 2.36·10-s − 1.66·11-s + 12-s − 13-s + 4.26·14-s + 2.36·15-s + 16-s + 1.26·17-s − 18-s − 4.50·19-s + 2.36·20-s − 4.26·21-s + 1.66·22-s + 8.71·23-s − 24-s + 0.580·25-s + 26-s + 27-s − 4.26·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.05·5-s − 0.408·6-s − 1.61·7-s − 0.353·8-s + 0.333·9-s − 0.746·10-s − 0.502·11-s + 0.288·12-s − 0.277·13-s + 1.13·14-s + 0.609·15-s + 0.250·16-s + 0.307·17-s − 0.235·18-s − 1.03·19-s + 0.528·20-s − 0.929·21-s + 0.355·22-s + 1.81·23-s − 0.204·24-s + 0.116·25-s + 0.196·26-s + 0.192·27-s − 0.805·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 2.36T + 5T^{2} \) |
| 7 | \( 1 + 4.26T + 7T^{2} \) |
| 11 | \( 1 + 1.66T + 11T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 23 | \( 1 - 8.71T + 23T^{2} \) |
| 29 | \( 1 - 7.70T + 29T^{2} \) |
| 31 | \( 1 + 7.02T + 31T^{2} \) |
| 37 | \( 1 - 1.42T + 37T^{2} \) |
| 41 | \( 1 + 2.88T + 41T^{2} \) |
| 43 | \( 1 + 2.87T + 43T^{2} \) |
| 47 | \( 1 - 2.32T + 47T^{2} \) |
| 53 | \( 1 + 3.14T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 4.27T + 61T^{2} \) |
| 67 | \( 1 + 9.62T + 67T^{2} \) |
| 71 | \( 1 + 7.37T + 71T^{2} \) |
| 73 | \( 1 + 5.92T + 73T^{2} \) |
| 79 | \( 1 - 5.42T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
| 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
−7.35061125294277680483254012075, −6.91345284210374282668694304399, −6.24979056765980520282641875256, −5.64297144781018032057047606365, −4.71907560699107651661473100262, −3.55647449882894083703299596162, −2.85598512775288158305526742246, −2.35586675860125948721941227045, −1.27542452568797168325844931754, 0,
1.27542452568797168325844931754, 2.35586675860125948721941227045, 2.85598512775288158305526742246, 3.55647449882894083703299596162, 4.71907560699107651661473100262, 5.64297144781018032057047606365, 6.24979056765980520282641875256, 6.91345284210374282668694304399, 7.35061125294277680483254012075
