| L(s) = 1 | − 2-s + 0.246·3-s + 4-s + 0.801·5-s − 0.246·6-s − 7-s − 8-s − 2.93·9-s − 0.801·10-s + 3.55·11-s + 0.246·12-s + 14-s + 0.198·15-s + 16-s − 1.22·17-s + 2.93·18-s + 2.75·19-s + 0.801·20-s − 0.246·21-s − 3.55·22-s + 2.29·23-s − 0.246·24-s − 4.35·25-s − 1.46·27-s − 28-s + 0.0217·29-s − 0.198·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.142·3-s + 0.5·4-s + 0.358·5-s − 0.100·6-s − 0.377·7-s − 0.353·8-s − 0.979·9-s − 0.253·10-s + 1.07·11-s + 0.0712·12-s + 0.267·14-s + 0.0511·15-s + 0.250·16-s − 0.297·17-s + 0.692·18-s + 0.631·19-s + 0.179·20-s − 0.0538·21-s − 0.757·22-s + 0.478·23-s − 0.0504·24-s − 0.871·25-s − 0.282·27-s − 0.188·28-s + 0.00404·29-s − 0.0361·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.301831127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.301831127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.246T + 3T^{2} \) |
| 5 | \( 1 - 0.801T + 5T^{2} \) |
| 11 | \( 1 - 3.55T + 11T^{2} \) |
| 17 | \( 1 + 1.22T + 17T^{2} \) |
| 19 | \( 1 - 2.75T + 19T^{2} \) |
| 23 | \( 1 - 2.29T + 23T^{2} \) |
| 29 | \( 1 - 0.0217T + 29T^{2} \) |
| 31 | \( 1 - 2.58T + 31T^{2} \) |
| 37 | \( 1 - 4.40T + 37T^{2} \) |
| 41 | \( 1 - 12.6T + 41T^{2} \) |
| 43 | \( 1 + 1.14T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 2.02T + 53T^{2} \) |
| 59 | \( 1 - 5.55T + 59T^{2} \) |
| 61 | \( 1 + 5.43T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 6.75T + 71T^{2} \) |
| 73 | \( 1 + 2.13T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 7.13T + 97T^{2} \) |
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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
−9.231608904192491586902712548891, −8.281523913998212875743497022512, −7.60551011148246263484782856015, −6.57143308921083238014837389108, −6.12699738140616982210206339496, −5.19865218274957874825417263739, −3.94745850858378560479600546269, −3.02860877027491343033506569454, −2.09571246210658478660392618411, −0.808789380427477110023012557831,
0.808789380427477110023012557831, 2.09571246210658478660392618411, 3.02860877027491343033506569454, 3.94745850858378560479600546269, 5.19865218274957874825417263739, 6.12699738140616982210206339496, 6.57143308921083238014837389108, 7.60551011148246263484782856015, 8.281523913998212875743497022512, 9.231608904192491586902712548891
