| L(s) = 1 | + 2.44·3-s + 15.1·5-s − 7·7-s − 21.0·9-s + 58.1·11-s − 71.5·13-s + 37.0·15-s − 90.9·17-s − 23.5·19-s − 17.1·21-s − 4.31·23-s + 104.·25-s − 117.·27-s − 148.·29-s + 69.4·31-s + 142.·33-s − 105.·35-s − 402.·37-s − 174.·39-s − 41·41-s − 242.·43-s − 318.·45-s + 72.3·47-s + 49·49-s − 222.·51-s − 10.2·53-s + 879.·55-s + ⋯ |
| L(s) = 1 | + 0.470·3-s + 1.35·5-s − 0.377·7-s − 0.778·9-s + 1.59·11-s − 1.52·13-s + 0.637·15-s − 1.29·17-s − 0.284·19-s − 0.177·21-s − 0.0391·23-s + 0.832·25-s − 0.837·27-s − 0.949·29-s + 0.402·31-s + 0.750·33-s − 0.511·35-s − 1.78·37-s − 0.718·39-s − 0.156·41-s − 0.858·43-s − 1.05·45-s + 0.224·47-s + 0.142·49-s − 0.611·51-s − 0.0265·53-s + 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 41 | \( 1 + 41T \) |
| good | 3 | \( 1 - 2.44T + 27T^{2} \) |
| 5 | \( 1 - 15.1T + 125T^{2} \) |
| 11 | \( 1 - 58.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 71.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 90.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 23.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 4.31T + 1.21e4T^{2} \) |
| 29 | \( 1 + 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 69.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 402.T + 5.06e4T^{2} \) |
| 43 | \( 1 + 242.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 72.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 10.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 188.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 625.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 624.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 391.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 51.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 944.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 937.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 621.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.61e3T + 9.12e5T^{2} \) |
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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
−9.161004471046655970931180592814, −8.510700528747340466346213026458, −7.13926661029786746295621741499, −6.52119462481137728622348877536, −5.71335285429017730208955245365, −4.73799012086297139476588888703, −3.55227428977262822954893506842, −2.42489138950159062105376076779, −1.76277358962492784015418681943, 0,
1.76277358962492784015418681943, 2.42489138950159062105376076779, 3.55227428977262822954893506842, 4.73799012086297139476588888703, 5.71335285429017730208955245365, 6.52119462481137728622348877536, 7.13926661029786746295621741499, 8.510700528747340466346213026458, 9.161004471046655970931180592814
