L(s) = 1 | − 2.56·3-s + 1.07·5-s − 1.61·7-s + 3.55·9-s − 0.256·11-s − 4.03·13-s − 2.74·15-s + 4.59·17-s + 3.87·19-s + 4.14·21-s − 2.28·23-s − 3.85·25-s − 1.42·27-s − 9.89·29-s + 7.10·31-s + 0.656·33-s − 1.73·35-s + 8.97·37-s + 10.3·39-s − 3.51·43-s + 3.81·45-s + 1.49·47-s − 4.38·49-s − 11.7·51-s − 12.7·53-s − 0.274·55-s − 9.91·57-s + ⋯ |
L(s) = 1 | − 1.47·3-s + 0.479·5-s − 0.611·7-s + 1.18·9-s − 0.0772·11-s − 1.11·13-s − 0.708·15-s + 1.11·17-s + 0.888·19-s + 0.903·21-s − 0.476·23-s − 0.770·25-s − 0.274·27-s − 1.83·29-s + 1.27·31-s + 0.114·33-s − 0.292·35-s + 1.47·37-s + 1.65·39-s − 0.536·43-s + 0.568·45-s + 0.218·47-s − 0.626·49-s − 1.64·51-s − 1.74·53-s − 0.0370·55-s − 1.31·57-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 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 1.07T + 5T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 0.256T + 11T^{2} \) |
| 13 | \( 1 + 4.03T + 13T^{2} \) |
| 17 | \( 1 - 4.59T + 17T^{2} \) |
| 19 | \( 1 - 3.87T + 19T^{2} \) |
| 23 | \( 1 + 2.28T + 23T^{2} \) |
| 29 | \( 1 + 9.89T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 - 8.97T + 37T^{2} \) |
| 43 | \( 1 + 3.51T + 43T^{2} \) |
| 47 | \( 1 - 1.49T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 + 0.543T + 61T^{2} \) |
| 67 | \( 1 - 4.41T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 4.68T + 79T^{2} \) |
| 83 | \( 1 - 0.168T + 83T^{2} \) |
| 89 | \( 1 + 4.75T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−7.55985195766559126240523294025, −6.70151583549973676913232772022, −6.16199657985703180743553392948, −5.45755547512602204465721611788, −5.13861502542778440261131836023, −4.14356855114520078614254067352, −3.20372825695772905472563334368, −2.18658099231979193741339815157, −1.02801487066940822742392611294, 0,
1.02801487066940822742392611294, 2.18658099231979193741339815157, 3.20372825695772905472563334368, 4.14356855114520078614254067352, 5.13861502542778440261131836023, 5.45755547512602204465721611788, 6.16199657985703180743553392948, 6.70151583549973676913232772022, 7.55985195766559126240523294025