| L(s) = 1 | − 2.52·3-s + 3.65·5-s − 0.974·7-s + 3.36·9-s − 5.14·11-s − 13-s − 9.21·15-s − 4.16·17-s + 7.70·19-s + 2.45·21-s − 4.54·23-s + 8.33·25-s − 0.915·27-s − 29-s + 5.25·31-s + 12.9·33-s − 3.55·35-s + 7.99·37-s + 2.52·39-s + 4.80·41-s − 8.32·43-s + 12.2·45-s + 1.86·47-s − 6.05·49-s + 10.5·51-s − 5.13·53-s − 18.7·55-s + ⋯ |
| L(s) = 1 | − 1.45·3-s + 1.63·5-s − 0.368·7-s + 1.12·9-s − 1.55·11-s − 0.277·13-s − 2.37·15-s − 1.01·17-s + 1.76·19-s + 0.536·21-s − 0.948·23-s + 1.66·25-s − 0.176·27-s − 0.185·29-s + 0.943·31-s + 2.25·33-s − 0.601·35-s + 1.31·37-s + 0.403·39-s + 0.750·41-s − 1.27·43-s + 1.83·45-s + 0.272·47-s − 0.864·49-s + 1.47·51-s − 0.704·53-s − 2.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.52T + 3T^{2} \) |
| 5 | \( 1 - 3.65T + 5T^{2} \) |
| 7 | \( 1 + 0.974T + 7T^{2} \) |
| 11 | \( 1 + 5.14T + 11T^{2} \) |
| 17 | \( 1 + 4.16T + 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 23 | \( 1 + 4.54T + 23T^{2} \) |
| 31 | \( 1 - 5.25T + 31T^{2} \) |
| 37 | \( 1 - 7.99T + 37T^{2} \) |
| 41 | \( 1 - 4.80T + 41T^{2} \) |
| 43 | \( 1 + 8.32T + 43T^{2} \) |
| 47 | \( 1 - 1.86T + 47T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 8.02T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 6.78T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 9.68T + 83T^{2} \) |
| 89 | \( 1 + 4.61T + 89T^{2} \) |
| 97 | \( 1 + 7.36T + 97T^{2} \) |
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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
−7.54127562156362656925435825892, −6.71417278771387633078285591782, −6.16468022265609474530723578229, −5.54041409411531364895936564590, −5.19325834408461260231527618261, −4.44188848775291079946911589070, −2.94710234800948422067273228789, −2.30524743659129641636960238985, −1.17890911071607434833805702009, 0,
1.17890911071607434833805702009, 2.30524743659129641636960238985, 2.94710234800948422067273228789, 4.44188848775291079946911589070, 5.19325834408461260231527618261, 5.54041409411531364895936564590, 6.16468022265609474530723578229, 6.71417278771387633078285591782, 7.54127562156362656925435825892
