| L(s) = 1 | − 2·2-s + 8.87·3-s + 4·4-s − 20.0·5-s − 17.7·6-s − 9.42·7-s − 8·8-s + 51.7·9-s + 40.0·10-s + 49.5·11-s + 35.4·12-s − 22.6·13-s + 18.8·14-s − 177.·15-s + 16·16-s − 32.3·17-s − 103.·18-s − 80.1·20-s − 83.6·21-s − 99.0·22-s − 0.309·23-s − 70.9·24-s + 276.·25-s + 45.2·26-s + 219.·27-s − 37.7·28-s − 42.0·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.70·3-s + 0.5·4-s − 1.79·5-s − 1.20·6-s − 0.508·7-s − 0.353·8-s + 1.91·9-s + 1.26·10-s + 1.35·11-s + 0.853·12-s − 0.483·13-s + 0.359·14-s − 3.06·15-s + 0.250·16-s − 0.461·17-s − 1.35·18-s − 0.896·20-s − 0.869·21-s − 0.959·22-s − 0.00280·23-s − 0.603·24-s + 2.21·25-s + 0.341·26-s + 1.56·27-s − 0.254·28-s − 0.268·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 8.87T + 27T^{2} \) |
| 5 | \( 1 + 20.0T + 125T^{2} \) |
| 7 | \( 1 + 9.42T + 343T^{2} \) |
| 11 | \( 1 - 49.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 32.3T + 4.91e3T^{2} \) |
| 23 | \( 1 + 0.309T + 1.21e4T^{2} \) |
| 29 | \( 1 + 42.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 1.87T + 2.97e4T^{2} \) |
| 37 | \( 1 + 331.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 47.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 382.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 122.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 93.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 550.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 527.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 644.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 611.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 407.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 405.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 774.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 880.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−9.237632819019715153898551529601, −8.710166780961906789533253787844, −8.034204795775809087100464907364, −7.25037076476179416346347196612, −6.68484498463486937437329535319, −4.49549846829586382267100601867, −3.63704697592377830808463561266, −3.03935703869932392733297606709, −1.59294441415367333366058125316, 0,
1.59294441415367333366058125316, 3.03935703869932392733297606709, 3.63704697592377830808463561266, 4.49549846829586382267100601867, 6.68484498463486937437329535319, 7.25037076476179416346347196612, 8.034204795775809087100464907364, 8.710166780961906789533253787844, 9.237632819019715153898551529601
