| L(s) = 1 | + 10.7·2-s − 23.7·3-s + 84.0·4-s + 25·5-s − 255.·6-s − 188.·7-s + 560.·8-s + 321.·9-s + 269.·10-s + 121·11-s − 1.99e3·12-s + 270.·13-s − 2.03e3·14-s − 593.·15-s + 3.35e3·16-s − 1.87e3·17-s + 3.46e3·18-s − 361·19-s + 2.10e3·20-s + 4.47e3·21-s + 1.30e3·22-s + 3.79e3·23-s − 1.33e4·24-s + 625·25-s + 2.91e3·26-s − 1.86e3·27-s − 1.58e4·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 1.52·3-s + 2.62·4-s + 0.447·5-s − 2.90·6-s − 1.45·7-s + 3.09·8-s + 1.32·9-s + 0.851·10-s + 0.301·11-s − 4.00·12-s + 0.444·13-s − 2.76·14-s − 0.681·15-s + 3.27·16-s − 1.57·17-s + 2.51·18-s − 0.229·19-s + 1.17·20-s + 2.21·21-s + 0.574·22-s + 1.49·23-s − 4.72·24-s + 0.200·25-s + 0.846·26-s − 0.491·27-s − 3.81·28-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 | \( 1 - 10.7T + 32T^{2} \) |
| 3 | \( 1 + 23.7T + 243T^{2} \) |
| 7 | \( 1 + 188.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 270.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.87e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.79e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 339.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.19e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.11e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.84e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.34e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.11e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.18e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.16e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.42e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.84e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.74e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.09e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.55e4T + 8.58e9T^{2} \) |
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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
−8.877491390750793443771713048664, −6.99747197618452799751405701747, −6.54311154080014376892472452247, −6.25763653207404189173377392104, −5.25839837363714236637820251210, −4.67688838303514174642304653418, −3.63526094523554064343879129719, −2.73049621771098833770696207402, −1.42851139116213982277045355991, 0,
1.42851139116213982277045355991, 2.73049621771098833770696207402, 3.63526094523554064343879129719, 4.67688838303514174642304653418, 5.25839837363714236637820251210, 6.25763653207404189173377392104, 6.54311154080014376892472452247, 6.99747197618452799751405701747, 8.877491390750793443771713048664
