| L(s) = 1 | − 2.78·3-s − 0.592·5-s − 7-s + 4.73·9-s − 2.21·11-s + 5.00·13-s + 1.64·15-s + 4.32·17-s − 2.38·19-s + 2.78·21-s + 1.67·23-s − 4.64·25-s − 4.82·27-s − 1.27·29-s + 2.49·31-s + 6.15·33-s + 0.592·35-s − 4.24·37-s − 13.9·39-s + 41-s − 8.15·43-s − 2.80·45-s + 1.65·47-s + 49-s − 12.0·51-s − 7.60·53-s + 1.31·55-s + ⋯ |
| L(s) = 1 | − 1.60·3-s − 0.265·5-s − 0.377·7-s + 1.57·9-s − 0.667·11-s + 1.38·13-s + 0.425·15-s + 1.04·17-s − 0.547·19-s + 0.606·21-s + 0.348·23-s − 0.929·25-s − 0.927·27-s − 0.236·29-s + 0.448·31-s + 1.07·33-s + 0.100·35-s − 0.697·37-s − 2.23·39-s + 0.156·41-s − 1.24·43-s − 0.418·45-s + 0.242·47-s + 0.142·49-s − 1.68·51-s − 1.04·53-s + 0.176·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 2.78T + 3T^{2} \) |
| 5 | \( 1 + 0.592T + 5T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 13 | \( 1 - 5.00T + 13T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 - 1.67T + 23T^{2} \) |
| 29 | \( 1 + 1.27T + 29T^{2} \) |
| 31 | \( 1 - 2.49T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 43 | \( 1 + 8.15T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 + 7.60T + 53T^{2} \) |
| 59 | \( 1 + 6.12T + 59T^{2} \) |
| 61 | \( 1 - 5.08T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 2.62T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 - 4.77T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 6.37T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−9.651848276248134853251806835072, −8.492984645427314151263814567560, −7.62383696023153298735126613389, −6.62575798706271388719909980405, −5.95967165021279489441086033902, −5.32067625999185495944569383974, −4.29043642749220374124490282993, −3.24518233323989573112435292222, −1.39605086295081559782390318399, 0,
1.39605086295081559782390318399, 3.24518233323989573112435292222, 4.29043642749220374124490282993, 5.32067625999185495944569383974, 5.95967165021279489441086033902, 6.62575798706271388719909980405, 7.62383696023153298735126613389, 8.492984645427314151263814567560, 9.651848276248134853251806835072
