| L(s) = 1 | − 8.07·2-s + 9·3-s + 33.2·4-s − 25·5-s − 72.7·6-s − 39.3·7-s − 10.2·8-s + 81·9-s + 201.·10-s + 121·11-s + 299.·12-s − 220.·13-s + 318.·14-s − 225·15-s − 981.·16-s − 200.·17-s − 654.·18-s + 350.·19-s − 831.·20-s − 354.·21-s − 977.·22-s + 1.38e3·23-s − 91.9·24-s + 625·25-s + 1.78e3·26-s + 729·27-s − 1.30e3·28-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 0.577·3-s + 1.03·4-s − 0.447·5-s − 0.824·6-s − 0.303·7-s − 0.0564·8-s + 0.333·9-s + 0.638·10-s + 0.301·11-s + 0.600·12-s − 0.362·13-s + 0.433·14-s − 0.258·15-s − 0.958·16-s − 0.168·17-s − 0.476·18-s + 0.222·19-s − 0.464·20-s − 0.175·21-s − 0.430·22-s + 0.545·23-s − 0.0325·24-s + 0.200·25-s + 0.517·26-s + 0.192·27-s − 0.315·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 8.07T + 32T^{2} \) |
| 7 | \( 1 + 39.3T + 1.68e4T^{2} \) |
| 13 | \( 1 + 220.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 200.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 350.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.38e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.06e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 527.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.20e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 563.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.15e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.84e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.37e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.99e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.61e3T + 8.58e9T^{2} \) |
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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
−11.11166507778935650298849369025, −10.12212587692076421456333593473, −9.276701086529893061591256878817, −8.464866503631378434479604307829, −7.55467052155738386203294011291, −6.62754375736756506334974633803, −4.62418724297109671954963949375, −3.04182088284471260804033926316, −1.47606669972464682093444461920, 0,
1.47606669972464682093444461920, 3.04182088284471260804033926316, 4.62418724297109671954963949375, 6.62754375736756506334974633803, 7.55467052155738386203294011291, 8.464866503631378434479604307829, 9.276701086529893061591256878817, 10.12212587692076421456333593473, 11.11166507778935650298849369025
