| L(s) = 1 | + 5-s − 11-s − 13-s − 3·17-s − 8·19-s − 5·23-s + 25-s − 6·29-s − 2·31-s + 7·37-s − 3·41-s − 2·43-s + 6·47-s − 6·53-s − 55-s + 5·59-s − 14·61-s − 65-s − 67-s − 9·73-s + 3·79-s − 6·83-s − 3·85-s + 3·89-s − 8·95-s + 97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s − 0.277·13-s − 0.727·17-s − 1.83·19-s − 1.04·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s + 1.15·37-s − 0.468·41-s − 0.304·43-s + 0.875·47-s − 0.824·53-s − 0.134·55-s + 0.650·59-s − 1.79·61-s − 0.124·65-s − 0.122·67-s − 1.05·73-s + 0.337·79-s − 0.658·83-s − 0.325·85-s + 0.317·89-s − 0.820·95-s + 0.101·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.39138999370157, −12.80728593062949, −12.72215438583841, −11.98669130767333, −11.55956909467143, −10.92946120796947, −10.66503613425464, −10.19721559963350, −9.650008602817406, −9.182118231558242, −8.758154464823210, −8.254200160605424, −7.711682385883729, −7.310520929595689, −6.571871023713697, −6.257753663361042, −5.820185728219468, −5.210841877653519, −4.621728695006645, −4.142435633384622, −3.730964926023926, −2.829595949247987, −2.366694137979311, −1.904476113367517, −1.289298906604685, 0, 0,
1.289298906604685, 1.904476113367517, 2.366694137979311, 2.829595949247987, 3.730964926023926, 4.142435633384622, 4.621728695006645, 5.210841877653519, 5.820185728219468, 6.257753663361042, 6.571871023713697, 7.310520929595689, 7.711682385883729, 8.254200160605424, 8.758154464823210, 9.182118231558242, 9.650008602817406, 10.19721559963350, 10.66503613425464, 10.92946120796947, 11.55956909467143, 11.98669130767333, 12.72215438583841, 12.80728593062949, 13.39138999370157
