| L(s) = 1 | + 3-s − 1.04·5-s + 9-s − 5.93·11-s − 0.888·13-s − 1.04·15-s − 4.51·17-s + 3.47·19-s − 1.93·23-s − 3.91·25-s + 27-s + 8.38·29-s − 5.13·31-s − 5.93·33-s − 5.65·37-s − 0.888·39-s − 8.39·41-s + 0.743·43-s − 1.04·45-s + 4.21·47-s − 4.51·51-s + 6.91·53-s + 6.18·55-s + 3.47·57-s + 5.43·59-s + 10.9·61-s + 0.926·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.466·5-s + 0.333·9-s − 1.78·11-s − 0.246·13-s − 0.269·15-s − 1.09·17-s + 0.797·19-s − 0.402·23-s − 0.782·25-s + 0.192·27-s + 1.55·29-s − 0.921·31-s − 1.03·33-s − 0.929·37-s − 0.142·39-s − 1.31·41-s + 0.113·43-s − 0.155·45-s + 0.615·47-s − 0.632·51-s + 0.949·53-s + 0.833·55-s + 0.460·57-s + 0.708·59-s + 1.40·61-s + 0.114·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.426481989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.426481989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.04T + 5T^{2} \) |
| 11 | \( 1 + 5.93T + 11T^{2} \) |
| 13 | \( 1 + 0.888T + 13T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 23 | \( 1 + 1.93T + 23T^{2} \) |
| 29 | \( 1 - 8.38T + 29T^{2} \) |
| 31 | \( 1 + 5.13T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 + 8.39T + 41T^{2} \) |
| 43 | \( 1 - 0.743T + 43T^{2} \) |
| 47 | \( 1 - 4.21T + 47T^{2} \) |
| 53 | \( 1 - 6.91T + 53T^{2} \) |
| 59 | \( 1 - 5.43T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 6.20T + 67T^{2} \) |
| 71 | \( 1 - 3.72T + 71T^{2} \) |
| 73 | \( 1 - 3.49T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 - 5.00T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.74782474974743271820824270137, −7.18039340501056453411615833830, −6.51720999945481796789388033616, −5.38695267708514710161032875363, −5.05451721450352533665628525183, −4.10732186472889012429019627679, −3.41586471188220590481493813226, −2.55963408058261167942453417435, −2.01242376277755888637261136670, −0.52441350944287243730775537613,
0.52441350944287243730775537613, 2.01242376277755888637261136670, 2.55963408058261167942453417435, 3.41586471188220590481493813226, 4.10732186472889012429019627679, 5.05451721450352533665628525183, 5.38695267708514710161032875363, 6.51720999945481796789388033616, 7.18039340501056453411615833830, 7.74782474974743271820824270137
