L(s) = 1 | + 1.05·2-s + 2.27·3-s − 0.889·4-s − 0.660·5-s + 2.40·6-s + 3.15·7-s − 3.04·8-s + 2.19·9-s − 0.696·10-s + 2.39·11-s − 2.02·12-s + 4.12·13-s + 3.32·14-s − 1.50·15-s − 1.42·16-s + 0.223·17-s + 2.30·18-s − 2.55·19-s + 0.587·20-s + 7.19·21-s + 2.51·22-s + 8.39·23-s − 6.93·24-s − 4.56·25-s + 4.35·26-s − 1.84·27-s − 2.80·28-s + ⋯ |
L(s) = 1 | + 0.745·2-s + 1.31·3-s − 0.444·4-s − 0.295·5-s + 0.980·6-s + 1.19·7-s − 1.07·8-s + 0.730·9-s − 0.220·10-s + 0.720·11-s − 0.585·12-s + 1.14·13-s + 0.888·14-s − 0.388·15-s − 0.357·16-s + 0.0543·17-s + 0.544·18-s − 0.586·19-s + 0.131·20-s + 1.56·21-s + 0.537·22-s + 1.75·23-s − 1.41·24-s − 0.912·25-s + 0.853·26-s − 0.354·27-s − 0.530·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.342463669\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.342463669\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 983 | \( 1 - T \) |
good | 2 | \( 1 - 1.05T + 2T^{2} \) |
| 3 | \( 1 - 2.27T + 3T^{2} \) |
| 5 | \( 1 + 0.660T + 5T^{2} \) |
| 7 | \( 1 - 3.15T + 7T^{2} \) |
| 11 | \( 1 - 2.39T + 11T^{2} \) |
| 13 | \( 1 - 4.12T + 13T^{2} \) |
| 17 | \( 1 - 0.223T + 17T^{2} \) |
| 19 | \( 1 + 2.55T + 19T^{2} \) |
| 23 | \( 1 - 8.39T + 23T^{2} \) |
| 29 | \( 1 - 5.32T + 29T^{2} \) |
| 31 | \( 1 + 8.46T + 31T^{2} \) |
| 37 | \( 1 - 9.00T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 + 0.428T + 43T^{2} \) |
| 47 | \( 1 + 9.60T + 47T^{2} \) |
| 53 | \( 1 + 6.65T + 53T^{2} \) |
| 59 | \( 1 - 9.50T + 59T^{2} \) |
| 61 | \( 1 + 6.35T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 9.94T + 71T^{2} \) |
| 73 | \( 1 + 3.59T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 9.35T + 97T^{2} \) |
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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
−9.683826904172161190884027781544, −8.919957946285284100583425814209, −8.448880571204551881989550536802, −7.77281588044116134307468711005, −6.55449581799082001689688453440, −5.44832856484579355246405552990, −4.41730188960701406847212448942, −3.79305841733673664137787366893, −2.88034291934486793191037056494, −1.46977678712610590285330131420,
1.46977678712610590285330131420, 2.88034291934486793191037056494, 3.79305841733673664137787366893, 4.41730188960701406847212448942, 5.44832856484579355246405552990, 6.55449581799082001689688453440, 7.77281588044116134307468711005, 8.448880571204551881989550536802, 8.919957946285284100583425814209, 9.683826904172161190884027781544