| L(s) = 1 | + 0.0547·3-s − 5-s + 3.87·7-s − 2.99·9-s + 3.25·11-s − 6.91·13-s − 0.0547·15-s + 0.379·17-s + 4.37·19-s + 0.212·21-s + 1.01·23-s + 25-s − 0.328·27-s − 3.37·29-s − 6.22·31-s + 0.178·33-s − 3.87·35-s − 0.791·37-s − 0.378·39-s + 2.36·41-s + 0.241·43-s + 2.99·45-s − 6.57·47-s + 7.99·49-s + 0.0208·51-s + 9.52·53-s − 3.25·55-s + ⋯ |
| L(s) = 1 | + 0.0316·3-s − 0.447·5-s + 1.46·7-s − 0.998·9-s + 0.980·11-s − 1.91·13-s − 0.0141·15-s + 0.0921·17-s + 1.00·19-s + 0.0462·21-s + 0.211·23-s + 0.200·25-s − 0.0632·27-s − 0.627·29-s − 1.11·31-s + 0.0310·33-s − 0.654·35-s − 0.130·37-s − 0.0606·39-s + 0.369·41-s + 0.0368·43-s + 0.446·45-s − 0.959·47-s + 1.14·49-s + 0.00291·51-s + 1.30·53-s − 0.438·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0547T + 3T^{2} \) |
| 7 | \( 1 - 3.87T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 + 6.91T + 13T^{2} \) |
| 17 | \( 1 - 0.379T + 17T^{2} \) |
| 19 | \( 1 - 4.37T + 19T^{2} \) |
| 23 | \( 1 - 1.01T + 23T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 + 6.22T + 31T^{2} \) |
| 37 | \( 1 + 0.791T + 37T^{2} \) |
| 41 | \( 1 - 2.36T + 41T^{2} \) |
| 43 | \( 1 - 0.241T + 43T^{2} \) |
| 47 | \( 1 + 6.57T + 47T^{2} \) |
| 53 | \( 1 - 9.52T + 53T^{2} \) |
| 59 | \( 1 - 1.22T + 59T^{2} \) |
| 61 | \( 1 + 9.03T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 1.03T + 73T^{2} \) |
| 79 | \( 1 - 1.42T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 0.256T + 89T^{2} \) |
| 97 | \( 1 - 4.05T + 97T^{2} \) |
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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
−7.55063401835874206597012630623, −7.07696309263927805614750531015, −5.97836455741742592793187300003, −5.21166649420376062515359871240, −4.82227355615941657361225794810, −3.96337510379914350410800905605, −3.07214250798099200930426321248, −2.20994468572385158393473208100, −1.31455535381201338664025949617, 0,
1.31455535381201338664025949617, 2.20994468572385158393473208100, 3.07214250798099200930426321248, 3.96337510379914350410800905605, 4.82227355615941657361225794810, 5.21166649420376062515359871240, 5.97836455741742592793187300003, 7.07696309263927805614750531015, 7.55063401835874206597012630623
