| L(s) = 1 | + 0.907·2-s + 3-s − 1.17·4-s + 3.19·5-s + 0.907·6-s + 3.56·7-s − 2.88·8-s + 9-s + 2.89·10-s − 3.27·11-s − 1.17·12-s + 5.58·13-s + 3.23·14-s + 3.19·15-s − 0.259·16-s + 0.907·18-s − 4.23·19-s − 3.75·20-s + 3.56·21-s − 2.97·22-s − 4.60·23-s − 2.88·24-s + 5.19·25-s + 5.06·26-s + 27-s − 4.19·28-s + 2.08·29-s + ⋯ |
| L(s) = 1 | + 0.641·2-s + 0.577·3-s − 0.588·4-s + 1.42·5-s + 0.370·6-s + 1.34·7-s − 1.01·8-s + 0.333·9-s + 0.915·10-s − 0.987·11-s − 0.339·12-s + 1.54·13-s + 0.863·14-s + 0.824·15-s − 0.0649·16-s + 0.213·18-s − 0.971·19-s − 0.840·20-s + 0.777·21-s − 0.633·22-s − 0.959·23-s − 0.588·24-s + 1.03·25-s + 0.993·26-s + 0.192·27-s − 0.792·28-s + 0.387·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.042232531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.042232531\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.907T + 2T^{2} \) |
| 5 | \( 1 - 3.19T + 5T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 + 3.27T + 11T^{2} \) |
| 13 | \( 1 - 5.58T + 13T^{2} \) |
| 19 | \( 1 + 4.23T + 19T^{2} \) |
| 23 | \( 1 + 4.60T + 23T^{2} \) |
| 29 | \( 1 - 2.08T + 29T^{2} \) |
| 31 | \( 1 - 0.448T + 31T^{2} \) |
| 37 | \( 1 + 0.742T + 37T^{2} \) |
| 41 | \( 1 + 4.49T + 41T^{2} \) |
| 43 | \( 1 - 6.10T + 43T^{2} \) |
| 47 | \( 1 + 2.26T + 47T^{2} \) |
| 53 | \( 1 - 7.55T + 53T^{2} \) |
| 59 | \( 1 + 2.83T + 59T^{2} \) |
| 61 | \( 1 + 3.91T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 3.64T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 5.63T + 79T^{2} \) |
| 83 | \( 1 + 3.92T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 + 0.828T + 97T^{2} \) |
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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
−10.24443321313563227406041956087, −9.160276773479815496321130652964, −8.528839339085800699080903969174, −7.910453843802022386769396538167, −6.33679786566951091526258207142, −5.66683878502573002598064107319, −4.84489250682567367306647871608, −3.92769057528520176240743761640, −2.58842529892019162046232543403, −1.55904695566808867345971275389,
1.55904695566808867345971275389, 2.58842529892019162046232543403, 3.92769057528520176240743761640, 4.84489250682567367306647871608, 5.66683878502573002598064107319, 6.33679786566951091526258207142, 7.910453843802022386769396538167, 8.528839339085800699080903969174, 9.160276773479815496321130652964, 10.24443321313563227406041956087
