| L(s) = 1 | + 1.73·3-s + (−1.73 + 1.41i)5-s − 2·7-s + 2.99·9-s − 5.65i·11-s + (−2.99 + 2.44i)15-s − 3.46·21-s + (0.999 − 4.89i)25-s + 5.19·27-s + 10.3·29-s − 4.89i·31-s − 9.79i·33-s + (3.46 − 2.82i)35-s + (−5.19 + 4.24i)45-s − 3·49-s + ⋯ |
| L(s) = 1 | + 1.00·3-s + (−0.774 + 0.632i)5-s − 0.755·7-s + 0.999·9-s − 1.70i·11-s + (−0.774 + 0.632i)15-s − 0.755·21-s + (0.199 − 0.979i)25-s + 1.00·27-s + 1.92·29-s − 0.879i·31-s − 1.70i·33-s + (0.585 − 0.478i)35-s + (−0.774 + 0.632i)45-s − 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.843688713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.843688713\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 + (1.73 - 1.41i)T \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 14.1iT - 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 14.6iT - 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 19.5iT - 97T^{2} \) |
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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
−8.920549235071937807253042297190, −8.273770961203366508379190840850, −7.75742768395748844069041936062, −6.66614943306303814469330464034, −6.26413560208654578874956689615, −4.85305643680807661590647879360, −3.68293288472333325949891105179, −3.30229561026618056145331876431, −2.44700592499417615241129398976, −0.65038828366471350638881262615,
1.26320998212036097764306542801, 2.51932474252877980214338675371, 3.43903398399164501404732018528, 4.40793185243720684648719215900, 4.86342543598987039630628157073, 6.38840250001930732454950530171, 7.20128926716196058554084091440, 7.70985864118406660735799913545, 8.640769820100521849704441824558, 9.166978431314989588625088730227
