| L(s) = 1 | + 9·3-s + 30.1·5-s + 135.·7-s + 81·9-s + 121·11-s + 609.·13-s + 271.·15-s + 641.·17-s + 1.93e3·19-s + 1.21e3·21-s − 2.15e3·23-s − 2.21e3·25-s + 729·27-s + 7.10e3·29-s − 9.71e3·31-s + 1.08e3·33-s + 4.06e3·35-s + 1.39e4·37-s + 5.48e3·39-s + 1.54e4·41-s − 1.70e4·43-s + 2.43e3·45-s + 4.86e3·47-s + 1.42e3·49-s + 5.77e3·51-s − 3.76e4·53-s + 3.64e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.538·5-s + 1.04·7-s + 0.333·9-s + 0.301·11-s + 1.00·13-s + 0.311·15-s + 0.538·17-s + 1.23·19-s + 0.601·21-s − 0.849·23-s − 0.709·25-s + 0.192·27-s + 1.56·29-s − 1.81·31-s + 0.174·33-s + 0.561·35-s + 1.68·37-s + 0.577·39-s + 1.43·41-s − 1.40·43-s + 0.179·45-s + 0.321·47-s + 0.0847·49-s + 0.310·51-s − 1.84·53-s + 0.162·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.094010705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.094010705\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 11 | \( 1 - 121T \) |
| good | 5 | \( 1 - 30.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 135.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 609.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 641.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.93e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.10e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.39e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.54e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.70e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.86e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.76e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.24e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.99e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.89e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.87e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.89e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.92e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.83e4T + 8.58e9T^{2} \) |
| 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
−9.886786500921794343177571253458, −9.232775793102143915842349383927, −8.167283018867717147122837080144, −7.66033193887975036698467713893, −6.32750407489289952979433287274, −5.43513301366463018991083923087, −4.30297148168487059083126605742, −3.23135342988234401366984852748, −1.91378094570149013642275769160, −1.08903662551888323251838607248,
1.08903662551888323251838607248, 1.91378094570149013642275769160, 3.23135342988234401366984852748, 4.30297148168487059083126605742, 5.43513301366463018991083923087, 6.32750407489289952979433287274, 7.66033193887975036698467713893, 8.167283018867717147122837080144, 9.232775793102143915842349383927, 9.886786500921794343177571253458
