| L(s) = 1 | − 3.51·2-s − 19.6·4-s − 88.3·5-s − 157.·7-s + 181.·8-s + 311.·10-s + 700.·13-s + 553.·14-s − 11.8·16-s − 831.·17-s + 764.·19-s + 1.73e3·20-s − 1.02e3·23-s + 4.68e3·25-s − 2.46e3·26-s + 3.08e3·28-s + 902.·29-s − 7.63e3·31-s − 5.77e3·32-s + 2.92e3·34-s + 1.39e4·35-s − 1.38e4·37-s − 2.69e3·38-s − 1.60e4·40-s + 1.74e4·41-s + 4.26e3·43-s + 3.62e3·46-s + ⋯ |
| L(s) = 1 | − 0.622·2-s − 0.612·4-s − 1.58·5-s − 1.21·7-s + 1.00·8-s + 0.983·10-s + 1.14·13-s + 0.755·14-s − 0.0115·16-s − 0.697·17-s + 0.486·19-s + 0.968·20-s − 0.405·23-s + 1.49·25-s − 0.715·26-s + 0.743·28-s + 0.199·29-s − 1.42·31-s − 0.996·32-s + 0.434·34-s + 1.91·35-s − 1.66·37-s − 0.302·38-s − 1.58·40-s + 1.61·41-s + 0.351·43-s + 0.252·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.06837962346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06837962346\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 3.51T + 32T^{2} \) |
| 5 | \( 1 + 88.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 157.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 700.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 831.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 764.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.02e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 902.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.63e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.38e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.74e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.26e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.95e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.65e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.13e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.01e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.31e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.47e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.09e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.36e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.79e4T + 8.58e9T^{2} \) |
| 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
−9.024098507232539117402568695687, −8.434703698355867714311864986631, −7.61578402767065846703468917751, −6.91821523043985095741698932152, −5.84329650519419073234541378112, −4.56214347527659237777687779673, −3.80586115735969575280587412819, −3.19252846939715397135654859911, −1.37395676105794566630609640458, −0.13384366472601920504430869747,
0.13384366472601920504430869747, 1.37395676105794566630609640458, 3.19252846939715397135654859911, 3.80586115735969575280587412819, 4.56214347527659237777687779673, 5.84329650519419073234541378112, 6.91821523043985095741698932152, 7.61578402767065846703468917751, 8.434703698355867714311864986631, 9.024098507232539117402568695687
