| L(s) = 1 | + 1.82·3-s − 1.95·5-s − 7-s + 0.338·9-s + 11-s − 13-s − 3.57·15-s + 1.53·17-s + 1.02·19-s − 1.82·21-s + 0.209·23-s − 1.17·25-s − 4.86·27-s + 4.01·29-s + 3.67·31-s + 1.82·33-s + 1.95·35-s − 0.347·37-s − 1.82·39-s − 5.62·41-s − 7.22·43-s − 0.661·45-s + 4.38·47-s + 49-s + 2.81·51-s − 9.15·53-s − 1.95·55-s + ⋯ |
| L(s) = 1 | + 1.05·3-s − 0.874·5-s − 0.377·7-s + 0.112·9-s + 0.301·11-s − 0.277·13-s − 0.922·15-s + 0.373·17-s + 0.235·19-s − 0.398·21-s + 0.0435·23-s − 0.234·25-s − 0.935·27-s + 0.745·29-s + 0.660·31-s + 0.318·33-s + 0.330·35-s − 0.0571·37-s − 0.292·39-s − 0.878·41-s − 1.10·43-s − 0.0986·45-s + 0.638·47-s + 0.142·49-s + 0.393·51-s − 1.25·53-s − 0.263·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.82T + 3T^{2} \) |
| 5 | \( 1 + 1.95T + 5T^{2} \) |
| 17 | \( 1 - 1.53T + 17T^{2} \) |
| 19 | \( 1 - 1.02T + 19T^{2} \) |
| 23 | \( 1 - 0.209T + 23T^{2} \) |
| 29 | \( 1 - 4.01T + 29T^{2} \) |
| 31 | \( 1 - 3.67T + 31T^{2} \) |
| 37 | \( 1 + 0.347T + 37T^{2} \) |
| 41 | \( 1 + 5.62T + 41T^{2} \) |
| 43 | \( 1 + 7.22T + 43T^{2} \) |
| 47 | \( 1 - 4.38T + 47T^{2} \) |
| 53 | \( 1 + 9.15T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 6.30T + 61T^{2} \) |
| 67 | \( 1 - 0.353T + 67T^{2} \) |
| 71 | \( 1 - 0.472T + 71T^{2} \) |
| 73 | \( 1 - 1.08T + 73T^{2} \) |
| 79 | \( 1 + 9.75T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 + 7.05T + 97T^{2} \) |
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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
−8.099486389871617991398544021318, −7.58039106235708459764730590239, −6.78096290751278203082772401295, −5.95005298285687925639825661631, −4.89090177087398263760832320003, −4.06513182784230762222586423799, −3.29323498573313191794390188494, −2.77402988893210592892743056964, −1.53620006246135482142359615155, 0,
1.53620006246135482142359615155, 2.77402988893210592892743056964, 3.29323498573313191794390188494, 4.06513182784230762222586423799, 4.89090177087398263760832320003, 5.95005298285687925639825661631, 6.78096290751278203082772401295, 7.58039106235708459764730590239, 8.099486389871617991398544021318
