| L(s) = 1 | + 2.63·3-s − 0.447·5-s + 0.0352·7-s + 3.93·9-s − 5.17·11-s + 3.84·13-s − 1.17·15-s + 0.854·17-s − 3.62·19-s + 0.0928·21-s − 4.86·23-s − 4.79·25-s + 2.46·27-s − 7.51·29-s + 0.445·31-s − 13.6·33-s − 0.0157·35-s − 5.65·37-s + 10.1·39-s + 2.01·41-s + 2.01·43-s − 1.76·45-s − 3.77·47-s − 6.99·49-s + 2.24·51-s − 4.58·53-s + 2.31·55-s + ⋯ |
| L(s) = 1 | + 1.52·3-s − 0.200·5-s + 0.0133·7-s + 1.31·9-s − 1.55·11-s + 1.06·13-s − 0.304·15-s + 0.207·17-s − 0.831·19-s + 0.0202·21-s − 1.01·23-s − 0.959·25-s + 0.474·27-s − 1.39·29-s + 0.0800·31-s − 2.37·33-s − 0.00266·35-s − 0.929·37-s + 1.62·39-s + 0.315·41-s + 0.307·43-s − 0.262·45-s − 0.550·47-s − 0.999·49-s + 0.314·51-s − 0.629·53-s + 0.312·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 2.63T + 3T^{2} \) |
| 5 | \( 1 + 0.447T + 5T^{2} \) |
| 7 | \( 1 - 0.0352T + 7T^{2} \) |
| 11 | \( 1 + 5.17T + 11T^{2} \) |
| 13 | \( 1 - 3.84T + 13T^{2} \) |
| 17 | \( 1 - 0.854T + 17T^{2} \) |
| 19 | \( 1 + 3.62T + 19T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 + 7.51T + 29T^{2} \) |
| 31 | \( 1 - 0.445T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 - 2.01T + 41T^{2} \) |
| 43 | \( 1 - 2.01T + 43T^{2} \) |
| 47 | \( 1 + 3.77T + 47T^{2} \) |
| 53 | \( 1 + 4.58T + 53T^{2} \) |
| 59 | \( 1 - 0.373T + 59T^{2} \) |
| 61 | \( 1 + 9.87T + 61T^{2} \) |
| 67 | \( 1 - 3.22T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 8.31T + 73T^{2} \) |
| 79 | \( 1 - 4.16T + 79T^{2} \) |
| 83 | \( 1 + 5.85T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 7.76T + 97T^{2} \) |
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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
−7.920606220395802786879922582700, −7.41525019377377925350845379375, −6.31225458442252702717591785621, −5.62299693287842759245823388197, −4.63991314344261976607089874110, −3.73986064475728445878915648793, −3.31765662874653230681882209205, −2.30769004352753796281986734191, −1.75616816134860404691075003910, 0,
1.75616816134860404691075003910, 2.30769004352753796281986734191, 3.31765662874653230681882209205, 3.73986064475728445878915648793, 4.63991314344261976607089874110, 5.62299693287842759245823388197, 6.31225458442252702717591785621, 7.41525019377377925350845379375, 7.920606220395802786879922582700
