| L(s) = 1 | + 35.4·2-s + 194.·3-s + 744.·4-s − 251.·5-s + 6.91e3·6-s − 1.05e4·7-s + 8.25e3·8-s + 1.83e4·9-s − 8.92e3·10-s + 3.92e4·11-s + 1.45e5·12-s + 2.08e4·13-s − 3.75e5·14-s − 4.90e4·15-s − 8.87e4·16-s + 6.49e5·18-s − 8.88e5·19-s − 1.87e5·20-s − 2.06e6·21-s + 1.39e6·22-s − 1.18e6·23-s + 1.60e6·24-s − 1.88e6·25-s + 7.40e5·26-s − 2.67e5·27-s − 7.89e6·28-s − 2.03e6·29-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 1.38·3-s + 1.45·4-s − 0.180·5-s + 2.17·6-s − 1.66·7-s + 0.712·8-s + 0.930·9-s − 0.282·10-s + 0.807·11-s + 2.02·12-s + 0.202·13-s − 2.61·14-s − 0.250·15-s − 0.338·16-s + 1.45·18-s − 1.56·19-s − 0.262·20-s − 2.31·21-s + 1.26·22-s − 0.884·23-s + 0.989·24-s − 0.967·25-s + 0.317·26-s − 0.0970·27-s − 2.42·28-s − 0.533·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 35.4T + 512T^{2} \) |
| 3 | \( 1 - 194.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 251.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.05e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.92e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.08e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 8.88e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.18e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.03e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.00e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.15e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.68e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.58e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.67e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.53e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.95e5T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.32e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.90e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.05e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.01e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.16e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.86e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.13e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.19e9T + 7.60e17T^{2} \) |
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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
−9.531911121969409362256409378834, −8.951829461484544800146808637513, −7.65583715429690202781618948755, −6.50295234761971164367316441185, −5.94517562531726364041607598014, −4.11370878994540825655177044304, −3.79620124071495626839566569493, −2.86227269034404209286545522772, −2.02975708794287912560821654483, 0,
2.02975708794287912560821654483, 2.86227269034404209286545522772, 3.79620124071495626839566569493, 4.11370878994540825655177044304, 5.94517562531726364041607598014, 6.50295234761971164367316441185, 7.65583715429690202781618948755, 8.951829461484544800146808637513, 9.531911121969409362256409378834
