| L(s) = 1 | + 5.41·2-s − 128.·3-s − 482.·4-s − 1.44e3·5-s − 696.·6-s + 9.44e3·7-s − 5.38e3·8-s − 3.14e3·9-s − 7.84e3·10-s − 5.33e4·11-s + 6.20e4·12-s + 5.04e4·13-s + 5.11e4·14-s + 1.86e5·15-s + 2.17e5·16-s − 1.70e4·18-s − 3.61e5·19-s + 6.99e5·20-s − 1.21e6·21-s − 2.88e5·22-s − 3.78e5·23-s + 6.92e5·24-s + 1.46e5·25-s + 2.73e5·26-s + 2.93e6·27-s − 4.55e6·28-s − 4.53e6·29-s + ⋯ |
| L(s) = 1 | + 0.239·2-s − 0.916·3-s − 0.942·4-s − 1.03·5-s − 0.219·6-s + 1.48·7-s − 0.464·8-s − 0.159·9-s − 0.248·10-s − 1.09·11-s + 0.864·12-s + 0.490·13-s + 0.355·14-s + 0.950·15-s + 0.831·16-s − 0.0382·18-s − 0.635·19-s + 0.977·20-s − 1.36·21-s − 0.262·22-s − 0.281·23-s + 0.426·24-s + 0.0749·25-s + 0.117·26-s + 1.06·27-s − 1.40·28-s − 1.19·29-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
| \(L(5)\) |
\(\approx\) |
\(0.1217755545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1217755545\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 5.41T + 512T^{2} \) |
| 3 | \( 1 + 128.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.44e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.44e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.33e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.04e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 3.61e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.78e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.53e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.59e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.33e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.88e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.12e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.82e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.10e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.66e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.24e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.73e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.28e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.38e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 6.51e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.90e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.64e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.69e8T + 7.60e17T^{2} \) |
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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
−10.74899131989648730034601848353, −9.114222899322469550759676964998, −8.141481553426137847542966545431, −7.65909881070010406679935164317, −5.95730918770157675535348818915, −5.11733618702326442041169747463, −4.49591992791473200579576062602, −3.40282392597662263011291316264, −1.63284985826375574167989332086, −0.15926175768030139552009914101,
0.15926175768030139552009914101, 1.63284985826375574167989332086, 3.40282392597662263011291316264, 4.49591992791473200579576062602, 5.11733618702326442041169747463, 5.95730918770157675535348818915, 7.65909881070010406679935164317, 8.141481553426137847542966545431, 9.114222899322469550759676964998, 10.74899131989648730034601848353
