| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 5.16·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s − 10.3·10-s − 11.1·11-s − 12·12-s + 13·13-s + 14·14-s − 15.4·15-s + 16·16-s + 43.6·17-s − 18·18-s − 21.6·19-s + 20.6·20-s + 21·21-s + 22.3·22-s − 100.·23-s + 24·24-s − 98.3·25-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 + 0.461·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.326·10-s − 0.306·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.266·15-s + 0.250·16-s + 0.623·17-s − 0.235·18-s − 0.261·19-s + 0.230·20-s + 0.218·21-s + 0.216·22-s − 0.909·23-s + 0.204·24-s − 0.786·25-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 | \( 1 - 5.16T + 125T^{2} \) |
| 11 | \( 1 + 11.1T + 1.33e3T^{2} \) |
| 17 | \( 1 - 43.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 21.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 54.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 189.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 35.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 30.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 252.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 503.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 711.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 893.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 214.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 553.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 865.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 670.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 777.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 155.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.26e3T + 9.12e5T^{2} \) |
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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
−10.16162666566356427364973218420, −9.191904885275320204319467397842, −8.195504931908031212000110497819, −7.27573434518352434716827646452, −6.21538541767549709654011224792, −5.63410263126931980590328239809, −4.20840671017503279672394598489, −2.75740524898681029013479309929, −1.40775451961327151591985274482, 0,
1.40775451961327151591985274482, 2.75740524898681029013479309929, 4.20840671017503279672394598489, 5.63410263126931980590328239809, 6.21538541767549709654011224792, 7.27573434518352434716827646452, 8.195504931908031212000110497819, 9.191904885275320204319467397842, 10.16162666566356427364973218420
