| L(s) = 1 | + 2.43·3-s + 1.40·5-s − 1.75·7-s + 2.93·9-s − 3.85·11-s − 13-s + 3.42·15-s + 1.71·17-s + 0.294·19-s − 4.26·21-s − 7.98·23-s − 3.02·25-s − 0.161·27-s − 29-s − 6.74·31-s − 9.37·33-s − 2.45·35-s + 11.0·37-s − 2.43·39-s − 8.08·41-s + 2.08·43-s + 4.12·45-s + 7.47·47-s − 3.93·49-s + 4.17·51-s − 1.85·53-s − 5.40·55-s + ⋯ |
| L(s) = 1 | + 1.40·3-s + 0.628·5-s − 0.661·7-s + 0.977·9-s − 1.16·11-s − 0.277·13-s + 0.883·15-s + 0.415·17-s + 0.0676·19-s − 0.930·21-s − 1.66·23-s − 0.605·25-s − 0.0310·27-s − 0.185·29-s − 1.21·31-s − 1.63·33-s − 0.415·35-s + 1.81·37-s − 0.390·39-s − 1.26·41-s + 0.317·43-s + 0.614·45-s + 1.09·47-s − 0.562·49-s + 0.584·51-s − 0.254·53-s − 0.729·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.43T + 3T^{2} \) |
| 5 | \( 1 - 1.40T + 5T^{2} \) |
| 7 | \( 1 + 1.75T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 17 | \( 1 - 1.71T + 17T^{2} \) |
| 19 | \( 1 - 0.294T + 19T^{2} \) |
| 23 | \( 1 + 7.98T + 23T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 8.08T + 41T^{2} \) |
| 43 | \( 1 - 2.08T + 43T^{2} \) |
| 47 | \( 1 - 7.47T + 47T^{2} \) |
| 53 | \( 1 + 1.85T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 5.56T + 61T^{2} \) |
| 67 | \( 1 + 8.07T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 9.78T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 6.34T + 83T^{2} \) |
| 89 | \( 1 + 3.82T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.64594274592868557012227206330, −7.45742780216384935967683248319, −6.12518474179714652939950115037, −5.79182848944122589536321979470, −4.72232438747580707928263034627, −3.79201511060358279457913786720, −3.10555923748797888532277213388, −2.39217746254944566657226248601, −1.76114062282575106089906489027, 0,
1.76114062282575106089906489027, 2.39217746254944566657226248601, 3.10555923748797888532277213388, 3.79201511060358279457913786720, 4.72232438747580707928263034627, 5.79182848944122589536321979470, 6.12518474179714652939950115037, 7.45742780216384935967683248319, 7.64594274592868557012227206330
