| L(s) = 1 | − 0.725·2-s − 31.4·4-s + 46.7·5-s + 49·7-s + 46.0·8-s − 33.8·10-s − 666.·11-s − 650.·13-s − 35.5·14-s + 973.·16-s − 1.18e3·17-s − 1.56e3·19-s − 1.47e3·20-s + 482.·22-s + 1.10e3·23-s − 939.·25-s + 471.·26-s − 1.54e3·28-s − 2.39e3·29-s − 2.04e3·31-s − 2.17e3·32-s + 860.·34-s + 2.29e3·35-s + 1.07e3·37-s + 1.13e3·38-s + 2.15e3·40-s − 1.09e3·41-s + ⋯ |
| L(s) = 1 | − 0.128·2-s − 0.983·4-s + 0.836·5-s + 0.377·7-s + 0.254·8-s − 0.107·10-s − 1.65·11-s − 1.06·13-s − 0.0484·14-s + 0.950·16-s − 0.996·17-s − 0.994·19-s − 0.822·20-s + 0.212·22-s + 0.433·23-s − 0.300·25-s + 0.136·26-s − 0.371·28-s − 0.529·29-s − 0.382·31-s − 0.376·32-s + 0.127·34-s + 0.316·35-s + 0.129·37-s + 0.127·38-s + 0.212·40-s − 0.102·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 2 | \( 1 + 0.725T + 32T^{2} \) |
| 5 | \( 1 - 46.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 666.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 650.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.56e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.09e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.65e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.29e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.51e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 675.T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.29e4T + 8.58e9T^{2} \) |
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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
−13.32859028945119021848845057503, −12.69993870607601881754929643237, −10.83053485270417759958028197589, −9.895454711119492587162477480760, −8.786663241086592585910118323794, −7.55016373329611929872680324318, −5.63164621465331210356143094652, −4.58694324331215130232623911361, −2.30881229721857315672749637921, 0,
2.30881229721857315672749637921, 4.58694324331215130232623911361, 5.63164621465331210356143094652, 7.55016373329611929872680324318, 8.786663241086592585910118323794, 9.895454711119492587162477480760, 10.83053485270417759958028197589, 12.69993870607601881754929643237, 13.32859028945119021848845057503
