| L(s) = 1 | − 0.372·3-s + 9.11·5-s − 8.86·9-s − 11·11-s − 3.39·15-s − 43.3·23-s + 58.1·25-s + 6.64·27-s + 61.5·31-s + 4.09·33-s − 47.8·37-s − 80.7·45-s + 50·47-s + 49·49-s − 70·53-s − 100.·55-s − 10.4·59-s + 94.5·67-s + 16.1·69-s + 109.·71-s − 21.6·75-s + 77.2·81-s − 80.7·89-s − 22.9·93-s + 98.9·97-s + 97.4·99-s − 190·103-s + ⋯ |
| L(s) = 1 | − 0.124·3-s + 1.82·5-s − 0.984·9-s − 11-s − 0.226·15-s − 1.88·23-s + 2.32·25-s + 0.246·27-s + 1.98·31-s + 0.124·33-s − 1.29·37-s − 1.79·45-s + 1.06·47-s + 0.999·49-s − 1.32·53-s − 1.82·55-s − 0.176·59-s + 1.41·67-s + 0.233·69-s + 1.54·71-s − 0.288·75-s + 0.954·81-s − 0.907·89-s − 0.246·93-s + 1.02·97-s + 0.984·99-s − 1.84·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.208860751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208860751\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 0.372T + 9T^{2} \) |
| 5 | \( 1 - 9.11T + 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 43.3T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 61.5T + 961T^{2} \) |
| 37 | \( 1 + 47.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 50T + 2.20e3T^{2} \) |
| 53 | \( 1 + 70T + 2.80e3T^{2} \) |
| 59 | \( 1 + 10.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 94.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 109.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + 80.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 98.9T + 9.40e3T^{2} \) |
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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
−15.73723905326535143414588276420, −14.11784891723833833991284189876, −13.66034475304249368140265244139, −12.26958080031858582140609623566, −10.63269194122655259378884262506, −9.740981200514768740423181003043, −8.336353446416877372022385657091, −6.28533988465134585907187362519, −5.32331015562240943534999407578, −2.43965958780976937201231522276,
2.43965958780976937201231522276, 5.32331015562240943534999407578, 6.28533988465134585907187362519, 8.336353446416877372022385657091, 9.740981200514768740423181003043, 10.63269194122655259378884262506, 12.26958080031858582140609623566, 13.66034475304249368140265244139, 14.11784891723833833991284189876, 15.73723905326535143414588276420
