| L(s) = 1 | − 2·2-s + 2·4-s − 5-s + 2·10-s + 3·13-s − 4·16-s − 4·17-s − 19-s − 2·20-s + 4·23-s + 25-s − 6·26-s − 8·29-s + 31-s + 8·32-s + 8·34-s + 7·37-s + 2·38-s − 6·41-s − 43-s − 8·46-s − 2·47-s − 2·50-s + 6·52-s − 4·53-s + 16·58-s + 8·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s + 0.632·10-s + 0.832·13-s − 16-s − 0.970·17-s − 0.229·19-s − 0.447·20-s + 0.834·23-s + 1/5·25-s − 1.17·26-s − 1.48·29-s + 0.179·31-s + 1.41·32-s + 1.37·34-s + 1.15·37-s + 0.324·38-s − 0.937·41-s − 0.152·43-s − 1.17·46-s − 0.291·47-s − 0.282·50-s + 0.832·52-s − 0.549·53-s + 2.10·58-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.01505775914466, −12.66040835393167, −11.72656846368675, −11.48087249103158, −11.00951243625112, −10.86499384552888, −10.08863047884984, −9.802113499362558, −9.200699083948629, −8.826032093964049, −8.458852059495451, −8.046100666945085, −7.488068458081231, −7.085157755534058, −6.548753271430418, −6.217882117616543, −5.345614836282980, −4.949797468330713, −4.180829891441404, −3.871913474915688, −3.161820771878510, −2.429006541702203, −1.935605561528075, −1.268788201225743, −0.6710494577772117, 0,
0.6710494577772117, 1.268788201225743, 1.935605561528075, 2.429006541702203, 3.161820771878510, 3.871913474915688, 4.180829891441404, 4.949797468330713, 5.345614836282980, 6.217882117616543, 6.548753271430418, 7.085157755534058, 7.488068458081231, 8.046100666945085, 8.458852059495451, 8.826032093964049, 9.200699083948629, 9.802113499362558, 10.08863047884984, 10.86499384552888, 11.00951243625112, 11.48087249103158, 11.72656846368675, 12.66040835393167, 13.01505775914466
