| L(s) = 1 | + 2.10·2-s + 1.69·3-s + 2.44·4-s + 0.492·5-s + 3.58·6-s + 7-s + 0.948·8-s − 0.115·9-s + 1.03·10-s + 4.16·12-s + 5.30·13-s + 2.10·14-s + 0.836·15-s − 2.89·16-s + 3.03·17-s − 0.242·18-s − 4.66·19-s + 1.20·20-s + 1.69·21-s − 5.63·23-s + 1.61·24-s − 4.75·25-s + 11.1·26-s − 5.29·27-s + 2.44·28-s + 6.92·29-s + 1.76·30-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 0.980·3-s + 1.22·4-s + 0.220·5-s + 1.46·6-s + 0.377·7-s + 0.335·8-s − 0.0383·9-s + 0.328·10-s + 1.20·12-s + 1.47·13-s + 0.563·14-s + 0.215·15-s − 0.724·16-s + 0.736·17-s − 0.0572·18-s − 1.07·19-s + 0.269·20-s + 0.370·21-s − 1.17·23-s + 0.328·24-s − 0.951·25-s + 2.19·26-s − 1.01·27-s + 0.462·28-s + 1.28·29-s + 0.322·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.721218863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.721218863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.10T + 2T^{2} \) |
| 3 | \( 1 - 1.69T + 3T^{2} \) |
| 5 | \( 1 - 0.492T + 5T^{2} \) |
| 13 | \( 1 - 5.30T + 13T^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 19 | \( 1 + 4.66T + 19T^{2} \) |
| 23 | \( 1 + 5.63T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 1.44T + 41T^{2} \) |
| 43 | \( 1 - 2.88T + 43T^{2} \) |
| 47 | \( 1 + 8.75T + 47T^{2} \) |
| 53 | \( 1 - 6.63T + 53T^{2} \) |
| 59 | \( 1 + 8.35T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 - 5.94T + 71T^{2} \) |
| 73 | \( 1 + 3.77T + 73T^{2} \) |
| 79 | \( 1 + 8.80T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 3.10T + 89T^{2} \) |
| 97 | \( 1 + 6.31T + 97T^{2} \) |
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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
−10.31547933096206258072153996108, −9.206870826449840996548610985992, −8.395031060607578309092886304347, −7.71691206872630508143904654777, −6.17526863683565563084765768429, −5.95508719256896577840141434774, −4.56990588749322220627546470205, −3.81293224357315526036252878655, −2.95764055646994524001046703680, −1.88590663494183730316942258244,
1.88590663494183730316942258244, 2.95764055646994524001046703680, 3.81293224357315526036252878655, 4.56990588749322220627546470205, 5.95508719256896577840141434774, 6.17526863683565563084765768429, 7.71691206872630508143904654777, 8.395031060607578309092886304347, 9.206870826449840996548610985992, 10.31547933096206258072153996108
