| L(s) = 1 | + 1.73·2-s − 2.37·3-s + 1.01·4-s − 2.63·5-s − 4.11·6-s − 2.01·7-s − 1.70·8-s + 2.62·9-s − 4.57·10-s − 3.39·11-s − 2.41·12-s + 5.63·13-s − 3.49·14-s + 6.24·15-s − 4.99·16-s + 0.866·17-s + 4.55·18-s + 2.46·19-s − 2.67·20-s + 4.76·21-s − 5.89·22-s − 6.37·23-s + 4.05·24-s + 1.92·25-s + 9.79·26-s + 0.892·27-s − 2.04·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 1.36·3-s + 0.508·4-s − 1.17·5-s − 1.68·6-s − 0.759·7-s − 0.603·8-s + 0.874·9-s − 1.44·10-s − 1.02·11-s − 0.695·12-s + 1.56·13-s − 0.933·14-s + 1.61·15-s − 1.24·16-s + 0.210·17-s + 1.07·18-s + 0.565·19-s − 0.598·20-s + 1.04·21-s − 1.25·22-s − 1.33·23-s + 0.826·24-s + 0.385·25-s + 1.91·26-s + 0.171·27-s − 0.386·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 241 | \( 1 + T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 + 2.63T + 5T^{2} \) |
| 7 | \( 1 + 2.01T + 7T^{2} \) |
| 11 | \( 1 + 3.39T + 11T^{2} \) |
| 13 | \( 1 - 5.63T + 13T^{2} \) |
| 17 | \( 1 - 0.866T + 17T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 + 6.37T + 23T^{2} \) |
| 29 | \( 1 + 4.52T + 29T^{2} \) |
| 31 | \( 1 + 3.51T + 31T^{2} \) |
| 37 | \( 1 + 5.19T + 37T^{2} \) |
| 41 | \( 1 - 1.35T + 41T^{2} \) |
| 43 | \( 1 - 8.49T + 43T^{2} \) |
| 47 | \( 1 + 9.44T + 47T^{2} \) |
| 53 | \( 1 - 9.71T + 53T^{2} \) |
| 59 | \( 1 - 6.03T + 59T^{2} \) |
| 61 | \( 1 - 4.45T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 3.01T + 71T^{2} \) |
| 73 | \( 1 + 0.255T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 0.273T + 97T^{2} \) |
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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
−11.68333929274102281886009065615, −11.24218908093756276081475172833, −10.13395786750348612302444683068, −8.569607541867734047161661583880, −7.26592642370704417203859589930, −6.04538947049761488988295190963, −5.51461354208593135315591798741, −4.21671258110227393819275669874, −3.35815133163609533258214494515, 0,
3.35815133163609533258214494515, 4.21671258110227393819275669874, 5.51461354208593135315591798741, 6.04538947049761488988295190963, 7.26592642370704417203859589930, 8.569607541867734047161661583880, 10.13395786750348612302444683068, 11.24218908093756276081475172833, 11.68333929274102281886009065615
