| L(s) = 1 | − 1.36·2-s + 2.51·3-s − 0.144·4-s + 5-s − 3.42·6-s − 7-s + 2.92·8-s + 3.32·9-s − 1.36·10-s − 0.362·11-s − 0.362·12-s + 1.06·13-s + 1.36·14-s + 2.51·15-s − 3.69·16-s + 1.33·17-s − 4.53·18-s + 6.02·19-s − 0.144·20-s − 2.51·21-s + 0.493·22-s − 23-s + 7.34·24-s + 25-s − 1.45·26-s + 0.827·27-s + 0.144·28-s + ⋯ |
| L(s) = 1 | − 0.963·2-s + 1.45·3-s − 0.0720·4-s + 0.447·5-s − 1.39·6-s − 0.377·7-s + 1.03·8-s + 1.10·9-s − 0.430·10-s − 0.109·11-s − 0.104·12-s + 0.295·13-s + 0.364·14-s + 0.649·15-s − 0.922·16-s + 0.322·17-s − 1.06·18-s + 1.38·19-s − 0.0322·20-s − 0.548·21-s + 0.105·22-s − 0.208·23-s + 1.49·24-s + 0.200·25-s − 0.284·26-s + 0.159·27-s + 0.0272·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.536627521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536627521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 3 | \( 1 - 2.51T + 3T^{2} \) |
| 11 | \( 1 + 0.362T + 11T^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 17 | \( 1 - 1.33T + 17T^{2} \) |
| 19 | \( 1 - 6.02T + 19T^{2} \) |
| 29 | \( 1 - 4.12T + 29T^{2} \) |
| 31 | \( 1 - 2.39T + 31T^{2} \) |
| 37 | \( 1 - 4.95T + 37T^{2} \) |
| 41 | \( 1 + 1.25T + 41T^{2} \) |
| 43 | \( 1 - 5.32T + 43T^{2} \) |
| 47 | \( 1 - 4.61T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 - 0.321T + 59T^{2} \) |
| 61 | \( 1 + 9.71T + 61T^{2} \) |
| 67 | \( 1 + 9.69T + 67T^{2} \) |
| 71 | \( 1 + 0.0407T + 71T^{2} \) |
| 73 | \( 1 + 9.48T + 73T^{2} \) |
| 79 | \( 1 + 6.10T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 6.43T + 97T^{2} \) |
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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
−9.917294133905672997931922421317, −9.267898537677611200769634604607, −8.775226712891470385258850099478, −7.85201817840030009286202279314, −7.36848249767167492777482740562, −6.03149325086905681086601953738, −4.70276803416894725065914555169, −3.52433296825927658189611310966, −2.53664469654007844435616302420, −1.21479807867403118552187471631,
1.21479807867403118552187471631, 2.53664469654007844435616302420, 3.52433296825927658189611310966, 4.70276803416894725065914555169, 6.03149325086905681086601953738, 7.36848249767167492777482740562, 7.85201817840030009286202279314, 8.775226712891470385258850099478, 9.267898537677611200769634604607, 9.917294133905672997931922421317
