| L(s) = 1 | + 5.78·2-s − 9·3-s + 1.45·4-s + 25·5-s − 52.0·6-s + 17.6·7-s − 176.·8-s + 81·9-s + 144.·10-s + 121·11-s − 13.1·12-s + 674.·13-s + 102.·14-s − 225·15-s − 1.06e3·16-s − 2.11e3·17-s + 468.·18-s − 2.30e3·19-s + 36.4·20-s − 158.·21-s + 699.·22-s − 3.07e3·23-s + 1.59e3·24-s + 625·25-s + 3.90e3·26-s − 729·27-s + 25.7·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s − 0.577·3-s + 0.0455·4-s + 0.447·5-s − 0.590·6-s + 0.136·7-s − 0.975·8-s + 0.333·9-s + 0.457·10-s + 0.301·11-s − 0.0262·12-s + 1.10·13-s + 0.139·14-s − 0.258·15-s − 1.04·16-s − 1.77·17-s + 0.340·18-s − 1.46·19-s + 0.0203·20-s − 0.0786·21-s + 0.308·22-s − 1.21·23-s + 0.563·24-s + 0.200·25-s + 1.13·26-s − 0.192·27-s + 0.00619·28-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 | \( 1 - 5.78T + 32T^{2} \) |
| 7 | \( 1 - 17.6T + 1.68e4T^{2} \) |
| 13 | \( 1 - 674.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.11e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.30e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.07e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.84e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.16e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.45e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.81e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.97e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 378.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.22e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.20e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.12e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.52e5T + 8.58e9T^{2} \) |
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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
−11.50276309236778477350397528295, −10.78224351647095496018801584601, −9.374430260570121437577126037473, −8.430857834540710248012987246452, −6.52482504148741178646838648247, −6.02222134005944540956859917722, −4.70271120020569756700175902464, −3.83698103902245781368943026269, −2.01705694368360652222090528519, 0,
2.01705694368360652222090528519, 3.83698103902245781368943026269, 4.70271120020569756700175902464, 6.02222134005944540956859917722, 6.52482504148741178646838648247, 8.430857834540710248012987246452, 9.374430260570121437577126037473, 10.78224351647095496018801584601, 11.50276309236778477350397528295
