| L(s) = 1 | − 1.36·3-s − 2.59·5-s − 4.39·7-s − 1.12·9-s − 3.29·11-s + 0.00728·13-s + 3.54·15-s − 1.33·17-s + 7.83·19-s + 6.01·21-s − 1.23·23-s + 1.72·25-s + 5.64·27-s − 6.60·29-s + 7.41·31-s + 4.50·33-s + 11.4·35-s + 7.46·37-s − 0.00996·39-s − 1.49·43-s + 2.92·45-s − 10.7·47-s + 12.3·49-s + 1.82·51-s + 13.1·53-s + 8.53·55-s − 10.7·57-s + ⋯ |
| L(s) = 1 | − 0.790·3-s − 1.15·5-s − 1.66·7-s − 0.375·9-s − 0.993·11-s + 0.00201·13-s + 0.915·15-s − 0.323·17-s + 1.79·19-s + 1.31·21-s − 0.258·23-s + 0.344·25-s + 1.08·27-s − 1.22·29-s + 1.33·31-s + 0.784·33-s + 1.92·35-s + 1.22·37-s − 0.00159·39-s − 0.227·43-s + 0.435·45-s − 1.57·47-s + 1.76·49-s + 0.255·51-s + 1.80·53-s + 1.15·55-s − 1.42·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 \) |
| good | 3 | \( 1 + 1.36T + 3T^{2} \) |
| 5 | \( 1 + 2.59T + 5T^{2} \) |
| 7 | \( 1 + 4.39T + 7T^{2} \) |
| 11 | \( 1 + 3.29T + 11T^{2} \) |
| 13 | \( 1 - 0.00728T + 13T^{2} \) |
| 17 | \( 1 + 1.33T + 17T^{2} \) |
| 19 | \( 1 - 7.83T + 19T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 29 | \( 1 + 6.60T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 - 7.46T + 37T^{2} \) |
| 43 | \( 1 + 1.49T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 3.33T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 4.07T + 67T^{2} \) |
| 71 | \( 1 + 5.39T + 71T^{2} \) |
| 73 | \( 1 + 8.42T + 73T^{2} \) |
| 79 | \( 1 - 7.92T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 8.14T + 89T^{2} \) |
| 97 | \( 1 - 6.29T + 97T^{2} \) |
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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
−7.57023868630879023854676838268, −6.92805131722827861925721457084, −6.17957613289158286374078677532, −5.58588640879659520407885348411, −4.87130069561644833816022931548, −3.88727294028046511078468078403, −3.21179390721699181387359383557, −2.60711959153125338836431484855, −0.75352498646814165098835994164, 0,
0.75352498646814165098835994164, 2.60711959153125338836431484855, 3.21179390721699181387359383557, 3.88727294028046511078468078403, 4.87130069561644833816022931548, 5.58588640879659520407885348411, 6.17957613289158286374078677532, 6.92805131722827861925721457084, 7.57023868630879023854676838268
