L(s) = 1 | − 2-s + 3-s + 4-s + 0.824·5-s − 6-s + 0.381·7-s − 8-s + 9-s − 0.824·10-s − 1.13·11-s + 12-s − 0.568·13-s − 0.381·14-s + 0.824·15-s + 16-s + 1.78·17-s − 18-s + 0.824·20-s + 0.381·21-s + 1.13·22-s + 9.04·23-s − 24-s − 4.32·25-s + 0.568·26-s + 27-s + 0.381·28-s − 3.50·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.368·5-s − 0.408·6-s + 0.144·7-s − 0.353·8-s + 0.333·9-s − 0.260·10-s − 0.343·11-s + 0.288·12-s − 0.157·13-s − 0.102·14-s + 0.212·15-s + 0.250·16-s + 0.432·17-s − 0.235·18-s + 0.184·20-s + 0.0833·21-s + 0.242·22-s + 1.88·23-s − 0.204·24-s − 0.864·25-s + 0.111·26-s + 0.192·27-s + 0.0721·28-s − 0.650·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.756406522\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.756406522\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 0.824T + 5T^{2} \) |
| 7 | \( 1 - 0.381T + 7T^{2} \) |
| 11 | \( 1 + 1.13T + 11T^{2} \) |
| 13 | \( 1 + 0.568T + 13T^{2} \) |
| 17 | \( 1 - 1.78T + 17T^{2} \) |
| 23 | \( 1 - 9.04T + 23T^{2} \) |
| 29 | \( 1 + 3.50T + 29T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 - 7.25T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 3.50T + 53T^{2} \) |
| 59 | \( 1 - 8.10T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 4.79T + 67T^{2} \) |
| 71 | \( 1 + 2.78T + 71T^{2} \) |
| 73 | \( 1 - 5.55T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 + 9.57T + 83T^{2} \) |
| 89 | \( 1 - 0.195T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.047048300850675382052300322121, −8.446150866378205155755533945618, −7.50375482608260012886449179421, −7.11384428450721886717102982870, −5.96601201918387107140771385452, −5.21275309314324409210030908929, −4.05346082768849311109966029488, −2.96522016979167393554027327616, −2.17242547531865990044072315463, −0.969628887864310013930170833353,
0.969628887864310013930170833353, 2.17242547531865990044072315463, 2.96522016979167393554027327616, 4.05346082768849311109966029488, 5.21275309314324409210030908929, 5.96601201918387107140771385452, 7.11384428450721886717102982870, 7.50375482608260012886449179421, 8.446150866378205155755533945618, 9.047048300850675382052300322121