| L(s) = 1 | − 3-s + 2.81·5-s − 7-s + 9-s − 1.71·11-s − 3.39·13-s − 2.81·15-s + 17-s − 0.813·19-s + 21-s − 2.28·23-s + 2.91·25-s − 27-s + 2.57·29-s + 7.10·31-s + 1.71·33-s − 2.81·35-s + 1.68·37-s + 3.39·39-s + 6.81·41-s − 9.54·43-s + 2.81·45-s − 10.3·47-s + 49-s − 51-s − 13.5·53-s − 4.81·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.25·5-s − 0.377·7-s + 0.333·9-s − 0.515·11-s − 0.940·13-s − 0.726·15-s + 0.242·17-s − 0.186·19-s + 0.218·21-s − 0.477·23-s + 0.583·25-s − 0.192·27-s + 0.478·29-s + 1.27·31-s + 0.297·33-s − 0.475·35-s + 0.276·37-s + 0.543·39-s + 1.06·41-s − 1.45·43-s + 0.419·45-s − 1.51·47-s + 0.142·49-s − 0.140·51-s − 1.85·53-s − 0.649·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2.81T + 5T^{2} \) |
| 11 | \( 1 + 1.71T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 19 | \( 1 + 0.813T + 19T^{2} \) |
| 23 | \( 1 + 2.28T + 23T^{2} \) |
| 29 | \( 1 - 2.57T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 - 1.68T + 37T^{2} \) |
| 41 | \( 1 - 6.81T + 41T^{2} \) |
| 43 | \( 1 + 9.54T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 - 1.94T + 59T^{2} \) |
| 61 | \( 1 - 3.30T + 61T^{2} \) |
| 67 | \( 1 - 8.88T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 2.52T + 79T^{2} \) |
| 83 | \( 1 + 8.57T + 83T^{2} \) |
| 89 | \( 1 + 8.78T + 89T^{2} \) |
| 97 | \( 1 + 2.57T + 97T^{2} \) |
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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
−7.80895205093501001009489460148, −6.71321098341773368822308155917, −6.43396806411405331990620981446, −5.56878897014391853736132462132, −5.06895161742215990518921671833, −4.27996679810269922628655217707, −3.02945840859105877272348453895, −2.33705098602204829746078684152, −1.36750029670696391613709312877, 0,
1.36750029670696391613709312877, 2.33705098602204829746078684152, 3.02945840859105877272348453895, 4.27996679810269922628655217707, 5.06895161742215990518921671833, 5.56878897014391853736132462132, 6.43396806411405331990620981446, 6.71321098341773368822308155917, 7.80895205093501001009489460148
