| L(s) = 1 | − 1.95·2-s − 3·3-s − 4.16·4-s − 18.6·5-s + 5.87·6-s − 1.49·7-s + 23.8·8-s + 9·9-s + 36.4·10-s − 35.5·11-s + 12.4·12-s − 46.8·13-s + 2.92·14-s + 55.8·15-s − 13.3·16-s + 77.7·17-s − 17.6·18-s + 38.9·19-s + 77.5·20-s + 4.48·21-s + 69.6·22-s − 23·23-s − 71.4·24-s + 221.·25-s + 91.7·26-s − 27·27-s + 6.22·28-s + ⋯ |
| L(s) = 1 | − 0.692·2-s − 0.577·3-s − 0.520·4-s − 1.66·5-s + 0.399·6-s − 0.0807·7-s + 1.05·8-s + 0.333·9-s + 1.15·10-s − 0.975·11-s + 0.300·12-s − 0.999·13-s + 0.0558·14-s + 0.961·15-s − 0.208·16-s + 1.10·17-s − 0.230·18-s + 0.470·19-s + 0.867·20-s + 0.0465·21-s + 0.675·22-s − 0.208·23-s − 0.607·24-s + 1.77·25-s + 0.692·26-s − 0.192·27-s + 0.0420·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0008206301317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0008206301317\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 23 | \( 1 + 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 1.95T + 8T^{2} \) |
| 5 | \( 1 + 18.6T + 125T^{2} \) |
| 7 | \( 1 + 1.49T + 343T^{2} \) |
| 11 | \( 1 + 35.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 77.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38.9T + 6.85e3T^{2} \) |
| 31 | \( 1 - 53.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 286.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 140.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 370.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 88.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 373.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 189.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 45.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 559.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 503.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 235.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 419.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 743.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.21e3T + 9.12e5T^{2} \) |
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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
−8.627137024715826929256292890026, −7.86407206359446692846770774255, −7.62317847455733354713919567412, −6.76536588007720045337776706604, −5.25029838699748495269800486535, −4.90989245726962071274830326312, −3.90348168539724111427682808512, −3.05415529203134527774026945262, −1.36003470212681676457914075220, −0.01646081721456742637124215498,
0.01646081721456742637124215498, 1.36003470212681676457914075220, 3.05415529203134527774026945262, 3.90348168539724111427682808512, 4.90989245726962071274830326312, 5.25029838699748495269800486535, 6.76536588007720045337776706604, 7.62317847455733354713919567412, 7.86407206359446692846770774255, 8.627137024715826929256292890026
