| L(s) = 1 | + 1.65·3-s − 4.30·5-s − 2.73·7-s − 0.270·9-s + 1.72·11-s + 0.546·13-s − 7.11·15-s − 3.82·17-s − 19-s − 4.51·21-s + 0.546·23-s + 13.5·25-s − 5.40·27-s + 0.0738·29-s − 1.49·31-s + 2.85·33-s + 11.7·35-s − 8.56·37-s + 0.902·39-s + 1.90·41-s − 9.07·43-s + 1.16·45-s − 5.33·47-s + 0.476·49-s − 6.32·51-s + 10.1·53-s − 7.44·55-s + ⋯ |
| L(s) = 1 | + 0.953·3-s − 1.92·5-s − 1.03·7-s − 0.0900·9-s + 0.521·11-s + 0.151·13-s − 1.83·15-s − 0.928·17-s − 0.229·19-s − 0.985·21-s + 0.113·23-s + 2.70·25-s − 1.03·27-s + 0.0137·29-s − 0.268·31-s + 0.497·33-s + 1.98·35-s − 1.40·37-s + 0.144·39-s + 0.297·41-s − 1.38·43-s + 0.173·45-s − 0.777·47-s + 0.0680·49-s − 0.885·51-s + 1.39·53-s − 1.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9382951572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9382951572\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.65T + 3T^{2} \) |
| 5 | \( 1 + 4.30T + 5T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 - 1.72T + 11T^{2} \) |
| 13 | \( 1 - 0.546T + 13T^{2} \) |
| 17 | \( 1 + 3.82T + 17T^{2} \) |
| 23 | \( 1 - 0.546T + 23T^{2} \) |
| 29 | \( 1 - 0.0738T + 29T^{2} \) |
| 31 | \( 1 + 1.49T + 31T^{2} \) |
| 37 | \( 1 + 8.56T + 37T^{2} \) |
| 41 | \( 1 - 1.90T + 41T^{2} \) |
| 43 | \( 1 + 9.07T + 43T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 6.03T + 59T^{2} \) |
| 61 | \( 1 - 1.31T + 61T^{2} \) |
| 67 | \( 1 - 9.02T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 8.50T + 73T^{2} \) |
| 79 | \( 1 - 7.68T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 7.30T + 97T^{2} \) |
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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
−8.372142822203576934617580528579, −7.65643715331373794700213195284, −6.90432714530960367138119145266, −6.46361790810554227666177937611, −5.15150915756721594901870186756, −4.22796125614675215306160486031, −3.51081552235427971402551761232, −3.27754000957605251337590660482, −2.13869186762723960090837838130, −0.48082096698678140942732121844,
0.48082096698678140942732121844, 2.13869186762723960090837838130, 3.27754000957605251337590660482, 3.51081552235427971402551761232, 4.22796125614675215306160486031, 5.15150915756721594901870186756, 6.46361790810554227666177937611, 6.90432714530960367138119145266, 7.65643715331373794700213195284, 8.372142822203576934617580528579
