| L(s) = 1 | − 8.15·2-s + 5.08·3-s + 34.4·4-s − 25·5-s − 41.4·6-s + 169.·7-s − 20.2·8-s − 217.·9-s + 203.·10-s − 121·11-s + 175.·12-s − 1.13e3·13-s − 1.38e3·14-s − 127.·15-s − 938.·16-s + 773.·17-s + 1.77e3·18-s − 361·19-s − 862.·20-s + 864.·21-s + 986.·22-s + 1.48e3·23-s − 103.·24-s + 625·25-s + 9.25e3·26-s − 2.33e3·27-s + 5.86e3·28-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 0.326·3-s + 1.07·4-s − 0.447·5-s − 0.470·6-s + 1.31·7-s − 0.112·8-s − 0.893·9-s + 0.644·10-s − 0.301·11-s + 0.351·12-s − 1.86·13-s − 1.88·14-s − 0.145·15-s − 0.916·16-s + 0.649·17-s + 1.28·18-s − 0.229·19-s − 0.481·20-s + 0.427·21-s + 0.434·22-s + 0.583·23-s − 0.0365·24-s + 0.200·25-s + 2.68·26-s − 0.617·27-s + 1.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 8.15T + 32T^{2} \) |
| 3 | \( 1 - 5.08T + 243T^{2} \) |
| 7 | \( 1 - 169.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.13e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 773.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.48e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.50e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.20e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.65e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.95e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.45e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.28e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.26e5T + 8.58e9T^{2} \) |
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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
−8.570200228988588343539778206393, −8.096570960981853843784173470458, −7.60281592253474278324913619532, −6.68047131456998643031433962802, −5.13235985443771244159663263936, −4.62214776425999592498084733319, −2.93769717776642653573263277827, −2.14645508803946091675640095472, −0.984598041156332945446260825940, 0,
0.984598041156332945446260825940, 2.14645508803946091675640095472, 2.93769717776642653573263277827, 4.62214776425999592498084733319, 5.13235985443771244159663263936, 6.68047131456998643031433962802, 7.60281592253474278324913619532, 8.096570960981853843784173470458, 8.570200228988588343539778206393
