| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 0.0134·5-s + 6·6-s − 2.50·7-s − 8·8-s + 9·9-s − 0.0268·10-s + 13.8·11-s − 12·12-s + 91.4·13-s + 5.00·14-s − 0.0403·15-s + 16·16-s + 30.9·17-s − 18·18-s + 0.0537·20-s + 7.51·21-s − 27.7·22-s − 108.·23-s + 24·24-s − 124.·25-s − 182.·26-s − 27·27-s − 10.0·28-s − 86.5·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.00120·5-s + 0.408·6-s − 0.135·7-s − 0.353·8-s + 0.333·9-s − 0.000850·10-s + 0.380·11-s − 0.288·12-s + 1.95·13-s + 0.0956·14-s − 0.000694·15-s + 0.250·16-s + 0.441·17-s − 0.235·18-s + 0.000601·20-s + 0.0780·21-s − 0.269·22-s − 0.984·23-s + 0.204·24-s − 0.999·25-s − 1.38·26-s − 0.192·27-s − 0.0676·28-s − 0.554·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 0.0134T + 125T^{2} \) |
| 7 | \( 1 + 2.50T + 343T^{2} \) |
| 11 | \( 1 - 13.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 91.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 30.9T + 4.91e3T^{2} \) |
| 23 | \( 1 + 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 86.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 144.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 288.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 155.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 11.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 89.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 569.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 121.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 637.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 472.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 904.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 676.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 496.T + 9.12e5T^{2} \) |
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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
−8.355810112802879588377733250662, −7.65826920785164388646606864284, −6.72321942545473647660152743977, −6.02554609229061775162456672501, −5.49264037589609960372328367740, −4.04097438768186851781564731292, −3.48163026311789759849914404996, −1.95428253773775227436209791143, −1.14853358853132070356134912503, 0,
1.14853358853132070356134912503, 1.95428253773775227436209791143, 3.48163026311789759849914404996, 4.04097438768186851781564731292, 5.49264037589609960372328367740, 6.02554609229061775162456672501, 6.72321942545473647660152743977, 7.65826920785164388646606864284, 8.355810112802879588377733250662
