| L(s) = 1 | + 2-s + 1.35·3-s + 4-s − 1.19·5-s + 1.35·6-s − 7-s + 8-s − 1.15·9-s − 1.19·10-s − 6.04·11-s + 1.35·12-s − 14-s − 1.62·15-s + 16-s + 6.82·17-s − 1.15·18-s − 6.85·19-s − 1.19·20-s − 1.35·21-s − 6.04·22-s + 2.69·23-s + 1.35·24-s − 3.56·25-s − 5.64·27-s − 28-s − 9.28·29-s − 1.62·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.783·3-s + 0.5·4-s − 0.535·5-s + 0.553·6-s − 0.377·7-s + 0.353·8-s − 0.386·9-s − 0.378·10-s − 1.82·11-s + 0.391·12-s − 0.267·14-s − 0.419·15-s + 0.250·16-s + 1.65·17-s − 0.273·18-s − 1.57·19-s − 0.267·20-s − 0.296·21-s − 1.28·22-s + 0.561·23-s + 0.276·24-s − 0.712·25-s − 1.08·27-s − 0.188·28-s − 1.72·29-s − 0.296·30-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.35T + 3T^{2} \) |
| 5 | \( 1 + 1.19T + 5T^{2} \) |
| 11 | \( 1 + 6.04T + 11T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 6.85T + 19T^{2} \) |
| 23 | \( 1 - 2.69T + 23T^{2} \) |
| 29 | \( 1 + 9.28T + 29T^{2} \) |
| 31 | \( 1 + 6.30T + 31T^{2} \) |
| 37 | \( 1 - 8.89T + 37T^{2} \) |
| 41 | \( 1 - 8.78T + 41T^{2} \) |
| 43 | \( 1 + 1.64T + 43T^{2} \) |
| 47 | \( 1 + 5.55T + 47T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 + 5.43T + 59T^{2} \) |
| 61 | \( 1 - 3.87T + 61T^{2} \) |
| 67 | \( 1 + 2.18T + 67T^{2} \) |
| 71 | \( 1 + 1.56T + 71T^{2} \) |
| 73 | \( 1 + 1.46T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 1.91T + 89T^{2} \) |
| 97 | \( 1 - 7.30T + 97T^{2} \) |
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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
−8.342531824041442860073419228907, −7.74624775585792899822370275045, −7.36328700216669049251654248319, −5.92001718552994453947585749829, −5.57169695986692731186114112425, −4.44071042888304606846456870165, −3.52749303112240668357871299744, −2.90894171740068516259659781615, −2.04799433319478496601540581539, 0,
2.04799433319478496601540581539, 2.90894171740068516259659781615, 3.52749303112240668357871299744, 4.44071042888304606846456870165, 5.57169695986692731186114112425, 5.92001718552994453947585749829, 7.36328700216669049251654248319, 7.74624775585792899822370275045, 8.342531824041442860073419228907
