L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 3.33·5-s + 6·6-s + 33.6·7-s − 8·8-s + 9·9-s + 6.67·10-s − 24.4·11-s − 12·12-s − 61.9·13-s − 67.3·14-s + 10.0·15-s + 16·16-s + 116.·17-s − 18·18-s − 13.3·20-s − 101.·21-s + 48.9·22-s + 111.·23-s + 24·24-s − 113.·25-s + 123.·26-s − 27·27-s + 134.·28-s − 108.·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.298·5-s + 0.408·6-s + 1.81·7-s − 0.353·8-s + 0.333·9-s + 0.211·10-s − 0.670·11-s − 0.288·12-s − 1.32·13-s − 1.28·14-s + 0.172·15-s + 0.250·16-s + 1.66·17-s − 0.235·18-s − 0.149·20-s − 1.04·21-s + 0.474·22-s + 1.01·23-s + 0.204·24-s − 0.910·25-s + 0.934·26-s − 0.192·27-s + 0.908·28-s − 0.696·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 3.33T + 125T^{2} \) |
| 7 | \( 1 - 33.6T + 343T^{2} \) |
| 11 | \( 1 + 24.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 116.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 145.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 134.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 71.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 247.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 423.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 582.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 165.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 692.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.03e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 542.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 667.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 235.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 640.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 833.T + 9.12e5T^{2} \) |
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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
−8.140098977756426006482657041591, −7.60053640766909212121090069143, −7.21087959626972080912986447536, −5.82123210115529456791263869338, −5.13600418453845719437946160546, −4.58296242720474701488963472735, −3.18402093245270885447859336835, −2.00430662519350173676914650356, −1.16778836752100155913287250477, 0,
1.16778836752100155913287250477, 2.00430662519350173676914650356, 3.18402093245270885447859336835, 4.58296242720474701488963472735, 5.13600418453845719437946160546, 5.82123210115529456791263869338, 7.21087959626972080912986447536, 7.60053640766909212121090069143, 8.140098977756426006482657041591