| L(s) = 1 | − 2.25·3-s − 4.27·5-s − 7-s + 2.08·9-s − 5.44·11-s + 9.64·15-s − 6.61·17-s + 2.49·19-s + 2.25·21-s + 5.96·23-s + 13.2·25-s + 2.05·27-s − 0.282·29-s + 1.74·31-s + 12.2·33-s + 4.27·35-s + 3.60·37-s − 3.74·41-s + 3.43·43-s − 8.93·45-s − 1.20·47-s + 49-s + 14.9·51-s + 8.16·53-s + 23.2·55-s − 5.63·57-s − 3.16·59-s + ⋯ |
| L(s) = 1 | − 1.30·3-s − 1.91·5-s − 0.377·7-s + 0.696·9-s − 1.64·11-s + 2.49·15-s − 1.60·17-s + 0.573·19-s + 0.492·21-s + 1.24·23-s + 2.65·25-s + 0.395·27-s − 0.0524·29-s + 0.313·31-s + 2.13·33-s + 0.722·35-s + 0.592·37-s − 0.585·41-s + 0.523·43-s − 1.33·45-s − 0.175·47-s + 0.142·49-s + 2.09·51-s + 1.12·53-s + 3.13·55-s − 0.746·57-s − 0.411·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.25T + 3T^{2} \) |
| 5 | \( 1 + 4.27T + 5T^{2} \) |
| 11 | \( 1 + 5.44T + 11T^{2} \) |
| 17 | \( 1 + 6.61T + 17T^{2} \) |
| 19 | \( 1 - 2.49T + 19T^{2} \) |
| 23 | \( 1 - 5.96T + 23T^{2} \) |
| 29 | \( 1 + 0.282T + 29T^{2} \) |
| 31 | \( 1 - 1.74T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 + 3.74T + 41T^{2} \) |
| 43 | \( 1 - 3.43T + 43T^{2} \) |
| 47 | \( 1 + 1.20T + 47T^{2} \) |
| 53 | \( 1 - 8.16T + 53T^{2} \) |
| 59 | \( 1 + 3.16T + 59T^{2} \) |
| 61 | \( 1 + 4.96T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 5.57T + 71T^{2} \) |
| 73 | \( 1 - 4.66T + 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 - 0.00717T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 + 3.20T + 97T^{2} \) |
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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
−7.77993467617956224259490231513, −7.17881114518498019854424860058, −6.61118831043964808513638033182, −5.65273112527185756122398564369, −4.80444443850808483107478611469, −4.51737423823296718291984516352, −3.37818971916665580215450447009, −2.63162730664729171674134737660, −0.73414796206686743725963938960, 0,
0.73414796206686743725963938960, 2.63162730664729171674134737660, 3.37818971916665580215450447009, 4.51737423823296718291984516352, 4.80444443850808483107478611469, 5.65273112527185756122398564369, 6.61118831043964808513638033182, 7.17881114518498019854424860058, 7.77993467617956224259490231513
