| L(s) = 1 | + (−0.618 + 1.90i)2-s − 2.35i·3-s + (−3.23 − 2.35i)4-s + (4.47 + 1.45i)6-s − 5.25i·7-s + (6.47 − 4.70i)8-s + 3.47·9-s − 19.9i·11-s + (−5.52 + 7.60i)12-s + 8.47·13-s + (9.99 + 3.24i)14-s + (4.94 + 15.2i)16-s − 11.8·17-s + (−2.14 + 6.60i)18-s + 15.2i·19-s + ⋯ |
| L(s) = 1 | + (−0.309 + 0.951i)2-s − 0.783i·3-s + (−0.809 − 0.587i)4-s + (0.745 + 0.242i)6-s − 0.751i·7-s + (0.809 − 0.587i)8-s + 0.385·9-s − 1.81i·11-s + (−0.460 + 0.634i)12-s + 0.651·13-s + (0.714 + 0.232i)14-s + (0.309 + 0.951i)16-s − 0.699·17-s + (−0.119 + 0.366i)18-s + 0.800i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.993313 - 0.322747i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.993313 - 0.322747i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.618 - 1.90i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.35iT - 9T^{2} \) |
| 7 | \( 1 + 5.25iT - 49T^{2} \) |
| 11 | \( 1 + 19.9iT - 121T^{2} \) |
| 13 | \( 1 - 8.47T + 169T^{2} \) |
| 17 | \( 1 + 11.8T + 289T^{2} \) |
| 19 | \( 1 - 15.2iT - 361T^{2} \) |
| 23 | \( 1 + 0.555iT - 529T^{2} \) |
| 29 | \( 1 + 10.9T + 841T^{2} \) |
| 31 | \( 1 - 8.29iT - 961T^{2} \) |
| 37 | \( 1 - 18.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 14.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 22.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 53.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 66.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 17.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 90.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 50.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 80.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.55T + 5.32e3T^{2} \) |
| 79 | \( 1 - 13.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 76.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 92.8T + 9.40e3T^{2} \) |
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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
−13.58144814031890766422283476548, −12.99632849313127240890040514415, −11.27541387699011730541714213852, −10.21716655462435373781634569368, −8.747264576796338528078542388035, −7.87269059786346726955626143986, −6.75713719630965874633579018260, −5.82071845095510883754967721621, −3.97154897075924117664650457065, −0.985298548415078959688056873277,
2.15929550937893682527185016823, 4.00428702953039914937341316926, 5.02836185335429757259945778664, 7.16457315494193985547599482270, 8.756763409779430242418521068584, 9.571938412786926647985624595110, 10.41862937294100602508350387807, 11.52481356374261451333981786654, 12.54417158411045161043191868825, 13.39446110409926090179597857186
