| L(s) = 1 | + 7-s − 3·11-s + 13-s − 7·17-s + 6·23-s + 5·29-s − 2·31-s + 2·37-s − 2·41-s + 4·43-s − 3·47-s + 49-s − 6·53-s + 10·59-s − 8·61-s − 2·67-s − 8·71-s + 6·73-s − 3·77-s + 5·79-s − 4·83-s + 91-s + 7·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.904·11-s + 0.277·13-s − 1.69·17-s + 1.25·23-s + 0.928·29-s − 0.359·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 0.437·47-s + 1/7·49-s − 0.824·53-s + 1.30·59-s − 1.02·61-s − 0.244·67-s − 0.949·71-s + 0.702·73-s − 0.341·77-s + 0.562·79-s − 0.439·83-s + 0.104·91-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.79052718717143, −15.05744093746535, −14.69533764371366, −13.92587745835059, −13.41511411712797, −13.04133108272807, −12.51128894513694, −11.77683258748116, −11.13328209127106, −10.87602335624557, −10.30632184517596, −9.564638619724319, −8.918365310689529, −8.536795814893093, −7.889415039618116, −7.255892463379500, −6.677508177179920, −6.097331951391058, −5.290139920655217, −4.758551668234514, −4.272025703868018, −3.316318578493567, −2.644395946982014, −2.004981694209399, −1.035997678551344, 0,
1.035997678551344, 2.004981694209399, 2.644395946982014, 3.316318578493567, 4.272025703868018, 4.758551668234514, 5.290139920655217, 6.097331951391058, 6.677508177179920, 7.255892463379500, 7.889415039618116, 8.536795814893093, 8.918365310689529, 9.564638619724319, 10.30632184517596, 10.87602335624557, 11.13328209127106, 11.77683258748116, 12.51128894513694, 13.04133108272807, 13.41511411712797, 13.92587745835059, 14.69533764371366, 15.05744093746535, 15.79052718717143
