L(s) = 1 | + (1.34 + 0.422i)2-s + 3-s + (1.64 + 1.14i)4-s + 1.97·5-s + (1.34 + 0.422i)6-s − 2.69·7-s + (1.73 + 2.23i)8-s + 9-s + (2.66 + 0.834i)10-s − 0.719i·11-s + (1.64 + 1.14i)12-s + 0.585i·13-s + (−3.63 − 1.13i)14-s + 1.97·15-s + (1.39 + 3.74i)16-s + 5.15i·17-s + ⋯ |
L(s) = 1 | + (0.954 + 0.299i)2-s + 0.577·3-s + (0.821 + 0.570i)4-s + 0.882·5-s + (0.550 + 0.172i)6-s − 1.01·7-s + (0.612 + 0.790i)8-s + 0.333·9-s + (0.842 + 0.263i)10-s − 0.217i·11-s + (0.474 + 0.329i)12-s + 0.162i·13-s + (−0.971 − 0.304i)14-s + 0.509·15-s + (0.348 + 0.937i)16-s + 1.24i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.05992 + 0.983892i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.05992 + 0.983892i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.34 - 0.422i)T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + (-0.180 + 4.79i)T \) |
good | 5 | \( 1 - 1.97T + 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 + 0.719iT - 11T^{2} \) |
| 13 | \( 1 - 0.585iT - 13T^{2} \) |
| 17 | \( 1 - 5.15iT - 17T^{2} \) |
| 19 | \( 1 + 6.29iT - 19T^{2} \) |
| 29 | \( 1 + 1.33iT - 29T^{2} \) |
| 31 | \( 1 - 0.359iT - 31T^{2} \) |
| 37 | \( 1 + 3.15T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 + 0.124iT - 43T^{2} \) |
| 47 | \( 1 + 7.12iT - 47T^{2} \) |
| 53 | \( 1 + 2.15T + 53T^{2} \) |
| 59 | \( 1 + 7.69T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 13.3iT - 67T^{2} \) |
| 71 | \( 1 + 8.35iT - 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 - 9.47T + 79T^{2} \) |
| 83 | \( 1 - 9.57iT - 83T^{2} \) |
| 89 | \( 1 + 13.9iT - 89T^{2} \) |
| 97 | \( 1 - 7.32iT - 97T^{2} \) |
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
−10.82180325280112381202913209187, −10.04610738519515176164788383623, −9.046202602230159981202582512786, −8.189968855572841129457844567700, −6.86289968029579744560851558768, −6.38432142079215240454501376827, −5.35742987182638741426562071165, −4.13124585044885452452633026638, −3.08151239298642687023795948800, −2.08072835638040879708049936268,
1.71082922870781533521905819652, 2.90370225165008101330815861384, 3.73755541647823011458360562029, 5.11791691643256159028208449875, 5.98997636732172972324551258717, 6.85127830145201001556405035677, 7.83335248256909875239797057236, 9.444204802438070202405345041453, 9.728433113890499768680775139052, 10.60448261970585827317737924055