| L(s) = 1 | + 3.54·2-s + 9·3-s − 19.4·4-s + 31.9·6-s + 15.4·7-s − 182.·8-s + 81·9-s + 121·11-s − 174.·12-s − 9.16·13-s + 54.6·14-s − 24.5·16-s − 1.48e3·17-s + 287.·18-s + 1.05e3·19-s + 138.·21-s + 428.·22-s − 3.00e3·23-s − 1.64e3·24-s − 32.4·26-s + 729·27-s − 299.·28-s + 3.76e3·29-s + 1.37e3·31-s + 5.74e3·32-s + 1.08e3·33-s − 5.25e3·34-s + ⋯ |
| L(s) = 1 | + 0.626·2-s + 0.577·3-s − 0.607·4-s + 0.361·6-s + 0.118·7-s − 1.00·8-s + 0.333·9-s + 0.301·11-s − 0.350·12-s − 0.0150·13-s + 0.0745·14-s − 0.0240·16-s − 1.24·17-s + 0.208·18-s + 0.672·19-s + 0.0686·21-s + 0.188·22-s − 1.18·23-s − 0.581·24-s − 0.00942·26-s + 0.192·27-s − 0.0722·28-s + 0.830·29-s + 0.257·31-s + 0.992·32-s + 0.174·33-s − 0.779·34-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
| \(L(3)\) |
\(\approx\) |
\(2.786425288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.786425288\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 3.54T + 32T^{2} \) |
| 7 | \( 1 - 15.4T + 1.68e4T^{2} \) |
| 13 | \( 1 + 9.16T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.48e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.05e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.00e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.37e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.01e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.65e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.36e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.63e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.71e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.91e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.22e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.82e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 419.T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.46e3T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.423850212965000263082947903237, −8.633313298471155619788487395763, −7.949642180101194044122132739264, −6.76582531766638273036145952931, −5.90016668652522914597234581248, −4.78719753753384144366969922312, −4.12977433589502825602704366809, −3.18765592732496765743122417918, −2.10786032024725582277639122027, −0.66430779616719198199258674748,
0.66430779616719198199258674748, 2.10786032024725582277639122027, 3.18765592732496765743122417918, 4.12977433589502825602704366809, 4.78719753753384144366969922312, 5.90016668652522914597234581248, 6.76582531766638273036145952931, 7.949642180101194044122132739264, 8.633313298471155619788487395763, 9.423850212965000263082947903237
