| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 7.62·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s + 15.2·10-s − 36.4·11-s − 12·12-s − 13·13-s − 14·14-s − 22.8·15-s + 16·16-s − 90.2·17-s + 18·18-s − 71.1·19-s + 30.4·20-s + 21·21-s − 72.9·22-s + 75.1·23-s − 24·24-s − 66.8·25-s − 26·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.681·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.482·10-s − 1.00·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.393·15-s + 0.250·16-s − 1.28·17-s + 0.235·18-s − 0.858·19-s + 0.340·20-s + 0.218·21-s − 0.707·22-s + 0.680·23-s − 0.204·24-s − 0.535·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 - 7.62T + 125T^{2} \) |
| 11 | \( 1 + 36.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 90.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 71.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 75.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 115.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 189.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 418.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 98.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 155.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 702.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 311.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 407.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 416.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 663.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 973.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 931.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 443.T + 9.12e5T^{2} \) |
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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
−10.30536951635691210989071048771, −9.201747600080390262344900604329, −8.050205721701552824871896088791, −6.80001894931559616168808531832, −6.25872739277498976758114174734, −5.20157917498516533829207657861, −4.48353441673830814066918005853, −2.98204653913667665445108772460, −1.89716309205247239044594928329, 0,
1.89716309205247239044594928329, 2.98204653913667665445108772460, 4.48353441673830814066918005853, 5.20157917498516533829207657861, 6.25872739277498976758114174734, 6.80001894931559616168808531832, 8.050205721701552824871896088791, 9.201747600080390262344900604329, 10.30536951635691210989071048771
