| L(s) = 1 | − 6.98·2-s − 9·3-s + 16.7·4-s + 62.8·6-s + 45.4·7-s + 106.·8-s + 81·9-s − 121·11-s − 150.·12-s − 398.·13-s − 317.·14-s − 1.27e3·16-s − 2.12e3·17-s − 565.·18-s − 1.80e3·19-s − 409.·21-s + 844.·22-s + 2.15e3·23-s − 957.·24-s + 2.77e3·26-s − 729·27-s + 762.·28-s + 7.07e3·29-s − 6.77e3·31-s + 5.52e3·32-s + 1.08e3·33-s + 1.48e4·34-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 0.577·3-s + 0.523·4-s + 0.712·6-s + 0.350·7-s + 0.587·8-s + 0.333·9-s − 0.301·11-s − 0.302·12-s − 0.653·13-s − 0.433·14-s − 1.24·16-s − 1.78·17-s − 0.411·18-s − 1.15·19-s − 0.202·21-s + 0.372·22-s + 0.848·23-s − 0.339·24-s + 0.806·26-s − 0.192·27-s + 0.183·28-s + 1.56·29-s − 1.26·31-s + 0.954·32-s + 0.174·33-s + 2.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3369637423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3369637423\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 6.98T + 32T^{2} \) |
| 7 | \( 1 - 45.4T + 1.68e4T^{2} \) |
| 13 | \( 1 + 398.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.12e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.80e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.07e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.77e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.35e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.08e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.10e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 687.T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.95e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.54e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.48e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.70e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.13e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.04e5T + 8.58e9T^{2} \) |
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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
−9.412073421774302788528303406980, −8.706682745966928432584669753501, −7.937484662155393162072460801677, −7.01749914600025116532247925397, −6.31723064405992671081048199049, −4.86997382798113166594579679151, −4.40160809775713598179981362221, −2.54859834654083152947643120455, −1.56206356172713759903741272305, −0.33374690207122717454888367296,
0.33374690207122717454888367296, 1.56206356172713759903741272305, 2.54859834654083152947643120455, 4.40160809775713598179981362221, 4.86997382798113166594579679151, 6.31723064405992671081048199049, 7.01749914600025116532247925397, 7.937484662155393162072460801677, 8.706682745966928432584669753501, 9.412073421774302788528303406980
