| L(s) = 1 | + 1.92·3-s + 1.09·5-s − 7-s + 0.714·9-s − 2.66·11-s + 2.10·15-s + 6.95·17-s − 5.13·19-s − 1.92·21-s − 1.06·23-s − 3.80·25-s − 4.40·27-s − 0.0985·29-s − 2.75·31-s − 5.14·33-s − 1.09·35-s + 4.39·37-s − 12.4·41-s + 8.19·43-s + 0.780·45-s + 7.46·47-s + 49-s + 13.4·51-s − 3.93·53-s − 2.91·55-s − 9.88·57-s + 0.410·59-s + ⋯ |
| L(s) = 1 | + 1.11·3-s + 0.488·5-s − 0.377·7-s + 0.238·9-s − 0.804·11-s + 0.543·15-s + 1.68·17-s − 1.17·19-s − 0.420·21-s − 0.221·23-s − 0.761·25-s − 0.847·27-s − 0.0183·29-s − 0.495·31-s − 0.894·33-s − 0.184·35-s + 0.721·37-s − 1.94·41-s + 1.25·43-s + 0.116·45-s + 1.08·47-s + 0.142·49-s + 1.87·51-s − 0.541·53-s − 0.392·55-s − 1.30·57-s + 0.0534·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.92T + 3T^{2} \) |
| 5 | \( 1 - 1.09T + 5T^{2} \) |
| 11 | \( 1 + 2.66T + 11T^{2} \) |
| 17 | \( 1 - 6.95T + 17T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 23 | \( 1 + 1.06T + 23T^{2} \) |
| 29 | \( 1 + 0.0985T + 29T^{2} \) |
| 31 | \( 1 + 2.75T + 31T^{2} \) |
| 37 | \( 1 - 4.39T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 8.19T + 43T^{2} \) |
| 47 | \( 1 - 7.46T + 47T^{2} \) |
| 53 | \( 1 + 3.93T + 53T^{2} \) |
| 59 | \( 1 - 0.410T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 - 0.197T + 67T^{2} \) |
| 71 | \( 1 + 3.61T + 71T^{2} \) |
| 73 | \( 1 + 9.10T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 2.39T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 7.53T + 97T^{2} \) |
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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
−7.68866656584585720494003741972, −6.73992962568083074021774540802, −5.82377120620278824441914846180, −5.53290380735417008118428125414, −4.41387319240245662936714979514, −3.63611037262863855517998655145, −2.96282316168378805609578850407, −2.33482742779720963552248250430, −1.50148556014775792624063500342, 0,
1.50148556014775792624063500342, 2.33482742779720963552248250430, 2.96282316168378805609578850407, 3.63611037262863855517998655145, 4.41387319240245662936714979514, 5.53290380735417008118428125414, 5.82377120620278824441914846180, 6.73992962568083074021774540802, 7.68866656584585720494003741972
