L(s) = 1 | + 2-s + 3-s + 4-s − 2.97·5-s + 6-s + 7-s + 8-s + 9-s − 2.97·10-s + 4.71·11-s + 12-s + 4.48·13-s + 14-s − 2.97·15-s + 16-s + 1.10·17-s + 18-s + 5.53·19-s − 2.97·20-s + 21-s + 4.71·22-s + 0.605·23-s + 24-s + 3.87·25-s + 4.48·26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.33·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.941·10-s + 1.42·11-s + 0.288·12-s + 1.24·13-s + 0.267·14-s − 0.769·15-s + 0.250·16-s + 0.268·17-s + 0.235·18-s + 1.27·19-s − 0.665·20-s + 0.218·21-s + 1.00·22-s + 0.126·23-s + 0.204·24-s + 0.774·25-s + 0.879·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.315006302\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.315006302\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
good | 5 | \( 1 + 2.97T + 5T^{2} \) |
| 11 | \( 1 - 4.71T + 11T^{2} \) |
| 13 | \( 1 - 4.48T + 13T^{2} \) |
| 17 | \( 1 - 1.10T + 17T^{2} \) |
| 19 | \( 1 - 5.53T + 19T^{2} \) |
| 23 | \( 1 - 0.605T + 23T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 31 | \( 1 - 2.50T + 31T^{2} \) |
| 37 | \( 1 - 3.25T + 37T^{2} \) |
| 41 | \( 1 - 8.59T + 41T^{2} \) |
| 43 | \( 1 + 3.30T + 43T^{2} \) |
| 47 | \( 1 + 9.81T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 2.94T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 2.14T + 67T^{2} \) |
| 71 | \( 1 + 3.79T + 71T^{2} \) |
| 73 | \( 1 + 6.45T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 5.26T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 6.68T + 97T^{2} \) |
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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
−7.79713649415974547041067180972, −7.24966807497697579447479993995, −6.43665138456470497300015429105, −5.80823416775685332783637042388, −4.69058095554482000414257305801, −4.23235930768759382584825520185, −3.40272142528162539969289236492, −3.24285584105587324954254493432, −1.73594671974393184558687890628, −0.983211542402108402158440735133,
0.983211542402108402158440735133, 1.73594671974393184558687890628, 3.24285584105587324954254493432, 3.40272142528162539969289236492, 4.23235930768759382584825520185, 4.69058095554482000414257305801, 5.80823416775685332783637042388, 6.43665138456470497300015429105, 7.24966807497697579447479993995, 7.79713649415974547041067180972