| L(s) = 1 | + 2.34·3-s − 3.88·5-s − 4.52·7-s + 2.49·9-s − 4.11·11-s − 5.22·13-s − 9.09·15-s − 17-s − 0.676·19-s − 10.5·21-s + 0.372·23-s + 10.0·25-s − 1.19·27-s − 10.3·29-s + 0.230·31-s − 9.64·33-s + 17.5·35-s − 9.93·37-s − 12.2·39-s + 0.729·41-s + 2.92·43-s − 9.67·45-s − 5.58·47-s + 13.4·49-s − 2.34·51-s + 6.39·53-s + 15.9·55-s + ⋯ |
| L(s) = 1 | + 1.35·3-s − 1.73·5-s − 1.70·7-s + 0.830·9-s − 1.24·11-s − 1.45·13-s − 2.34·15-s − 0.242·17-s − 0.155·19-s − 2.31·21-s + 0.0777·23-s + 2.01·25-s − 0.229·27-s − 1.92·29-s + 0.0414·31-s − 1.67·33-s + 2.96·35-s − 1.63·37-s − 1.96·39-s + 0.113·41-s + 0.446·43-s − 1.44·45-s − 0.814·47-s + 1.92·49-s − 0.328·51-s + 0.878·53-s + 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1711292507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1711292507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 2.34T + 3T^{2} \) |
| 5 | \( 1 + 3.88T + 5T^{2} \) |
| 7 | \( 1 + 4.52T + 7T^{2} \) |
| 11 | \( 1 + 4.11T + 11T^{2} \) |
| 13 | \( 1 + 5.22T + 13T^{2} \) |
| 19 | \( 1 + 0.676T + 19T^{2} \) |
| 23 | \( 1 - 0.372T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 0.230T + 31T^{2} \) |
| 37 | \( 1 + 9.93T + 37T^{2} \) |
| 41 | \( 1 - 0.729T + 41T^{2} \) |
| 43 | \( 1 - 2.92T + 43T^{2} \) |
| 47 | \( 1 + 5.58T + 47T^{2} \) |
| 53 | \( 1 - 6.39T + 53T^{2} \) |
| 61 | \( 1 - 6.42T + 61T^{2} \) |
| 67 | \( 1 - 0.909T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 4.28T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 6.69T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 8.33T + 97T^{2} \) |
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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
−7.73257194698114732010526408357, −7.30651107230280649882922668526, −6.93993788882218608664466291711, −5.70664626291291498282585815475, −4.83618168786200865282003134000, −3.91711737469466645477195178407, −3.44686106783624918244545026724, −2.86299605899804172937047256156, −2.21711244228217479579799333227, −0.17226823533249381482437713093,
0.17226823533249381482437713093, 2.21711244228217479579799333227, 2.86299605899804172937047256156, 3.44686106783624918244545026724, 3.91711737469466645477195178407, 4.83618168786200865282003134000, 5.70664626291291498282585815475, 6.93993788882218608664466291711, 7.30651107230280649882922668526, 7.73257194698114732010526408357
