| L(s) = 1 | + 5-s − 0.845·7-s + 4.55·11-s + 5.23·13-s − 4.72·17-s − 5.54·19-s − 23-s + 25-s − 7.06·29-s − 1.83·31-s − 0.845·35-s − 9.93·37-s − 5.71·41-s + 4.78·43-s − 5.54·47-s − 6.28·49-s + 14.4·53-s + 4.55·55-s − 5.06·59-s − 6.22·61-s + 5.23·65-s − 7.85·67-s − 3.71·71-s + 1.23·73-s − 3.85·77-s − 13.7·79-s − 7.50·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.319·7-s + 1.37·11-s + 1.45·13-s − 1.14·17-s − 1.27·19-s − 0.208·23-s + 0.200·25-s − 1.31·29-s − 0.329·31-s − 0.142·35-s − 1.63·37-s − 0.892·41-s + 0.729·43-s − 0.808·47-s − 0.897·49-s + 1.99·53-s + 0.614·55-s − 0.660·59-s − 0.796·61-s + 0.649·65-s − 0.960·67-s − 0.440·71-s + 0.144·73-s − 0.439·77-s − 1.54·79-s − 0.824·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 0.845T + 7T^{2} \) |
| 11 | \( 1 - 4.55T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 + 4.72T + 17T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 29 | \( 1 + 7.06T + 29T^{2} \) |
| 31 | \( 1 + 1.83T + 31T^{2} \) |
| 37 | \( 1 + 9.93T + 37T^{2} \) |
| 41 | \( 1 + 5.71T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 + 5.54T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 + 5.06T + 59T^{2} \) |
| 61 | \( 1 + 6.22T + 61T^{2} \) |
| 67 | \( 1 + 7.85T + 67T^{2} \) |
| 71 | \( 1 + 3.71T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 7.50T + 83T^{2} \) |
| 89 | \( 1 + 2.98T + 89T^{2} \) |
| 97 | \( 1 + 2.33T + 97T^{2} \) |
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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
−7.24787303145428331997538389264, −6.65852781578630321954787337055, −6.18102268845574781720426677124, −5.58516148744577378532974620790, −4.45375073071408578638415969709, −3.91890786403089708644430837746, −3.22892147656683882581059240387, −1.98113897031378407103206719384, −1.47276135087689712586132679436, 0,
1.47276135087689712586132679436, 1.98113897031378407103206719384, 3.22892147656683882581059240387, 3.91890786403089708644430837746, 4.45375073071408578638415969709, 5.58516148744577378532974620790, 6.18102268845574781720426677124, 6.65852781578630321954787337055, 7.24787303145428331997538389264
