| L(s) = 1 | − 34·7-s + 18·11-s − 12·13-s + 106·17-s − 44·19-s − 56·23-s + 270·29-s + 204·31-s − 120·37-s + 80·41-s − 536·43-s + 536·47-s + 813·49-s − 542·53-s − 174·59-s + 186·61-s − 332·67-s − 132·71-s + 602·73-s − 612·77-s − 548·79-s + 492·83-s − 1.05e3·89-s + 408·91-s − 482·97-s + 1.21e3·101-s − 898·103-s + ⋯ |
| L(s) = 1 | − 1.83·7-s + 0.493·11-s − 0.256·13-s + 1.51·17-s − 0.531·19-s − 0.507·23-s + 1.72·29-s + 1.18·31-s − 0.533·37-s + 0.304·41-s − 1.90·43-s + 1.66·47-s + 2.37·49-s − 1.40·53-s − 0.383·59-s + 0.390·61-s − 0.605·67-s − 0.220·71-s + 0.965·73-s − 0.905·77-s − 0.780·79-s + 0.650·83-s − 1.25·89-s + 0.470·91-s − 0.504·97-s + 1.19·101-s − 0.859·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 34 T + p^{3} T^{2} \) |
| 11 | \( 1 - 18 T + p^{3} T^{2} \) |
| 13 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 106 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 + 56 T + p^{3} T^{2} \) |
| 29 | \( 1 - 270 T + p^{3} T^{2} \) |
| 31 | \( 1 - 204 T + p^{3} T^{2} \) |
| 37 | \( 1 + 120 T + p^{3} T^{2} \) |
| 41 | \( 1 - 80 T + p^{3} T^{2} \) |
| 43 | \( 1 + 536 T + p^{3} T^{2} \) |
| 47 | \( 1 - 536 T + p^{3} T^{2} \) |
| 53 | \( 1 + 542 T + p^{3} T^{2} \) |
| 59 | \( 1 + 174 T + p^{3} T^{2} \) |
| 61 | \( 1 - 186 T + p^{3} T^{2} \) |
| 67 | \( 1 + 332 T + p^{3} T^{2} \) |
| 71 | \( 1 + 132 T + p^{3} T^{2} \) |
| 73 | \( 1 - 602 T + p^{3} T^{2} \) |
| 79 | \( 1 + 548 T + p^{3} T^{2} \) |
| 83 | \( 1 - 492 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1052 T + p^{3} T^{2} \) |
| 97 | \( 1 + 482 T + p^{3} T^{2} \) |
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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
−8.584232553168694314021033728585, −7.72876181082921071888120306898, −6.65113326413272609196858278755, −6.36475773943266278649277295625, −5.38424602421298264458701189354, −4.22572386700575919813598731174, −3.33768781813259207778091142683, −2.67389012941901964485779336576, −1.13503637543123197955099945814, 0,
1.13503637543123197955099945814, 2.67389012941901964485779336576, 3.33768781813259207778091142683, 4.22572386700575919813598731174, 5.38424602421298264458701189354, 6.36475773943266278649277295625, 6.65113326413272609196858278755, 7.72876181082921071888120306898, 8.584232553168694314021033728585
