| L(s) = 1 | + 31.1·2-s + 218.·3-s + 460.·4-s + 1.54e3·5-s + 6.79e3·6-s − 9.98e3·7-s − 1.61e3·8-s + 2.78e4·9-s + 4.81e4·10-s − 8.37e4·11-s + 1.00e5·12-s − 1.39e5·13-s − 3.11e5·14-s + 3.37e5·15-s − 2.86e5·16-s + 8.69e5·18-s + 2.96e5·19-s + 7.11e5·20-s − 2.17e6·21-s − 2.61e6·22-s + 1.40e6·23-s − 3.52e5·24-s + 4.34e5·25-s − 4.34e6·26-s + 1.78e6·27-s − 4.59e6·28-s + 2.36e6·29-s + ⋯ |
| L(s) = 1 | + 1.37·2-s + 1.55·3-s + 0.898·4-s + 1.10·5-s + 2.14·6-s − 1.57·7-s − 0.139·8-s + 1.41·9-s + 1.52·10-s − 1.72·11-s + 1.39·12-s − 1.35·13-s − 2.16·14-s + 1.71·15-s − 1.09·16-s + 1.95·18-s + 0.522·19-s + 0.993·20-s − 2.44·21-s − 2.37·22-s + 1.04·23-s − 0.216·24-s + 0.222·25-s − 1.86·26-s + 0.647·27-s − 1.41·28-s + 0.619·29-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 | \( 1 - 31.1T + 512T^{2} \) |
| 3 | \( 1 - 218.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.54e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 9.98e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.37e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.39e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 2.96e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.40e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.36e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.48e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.66e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.21e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.04e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.26e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.81e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.91e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.72e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 5.63e6T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.00e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.29e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 6.25e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.72e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.11e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.54e8T + 7.60e17T^{2} \) |
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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
−9.681055261817146656314256556735, −9.054761893608649584925645687993, −7.65894560863616799929739185189, −6.69744204209736132933317064150, −5.59536333486704019028419010487, −4.72599915027990531116827068314, −3.17589500968638196582557403716, −2.90830078601387950196340154452, −2.14191598568454575494967877300, 0,
2.14191598568454575494967877300, 2.90830078601387950196340154452, 3.17589500968638196582557403716, 4.72599915027990531116827068314, 5.59536333486704019028419010487, 6.69744204209736132933317064150, 7.65894560863616799929739185189, 9.054761893608649584925645687993, 9.681055261817146656314256556735
