| L(s) = 1 | − 3.08·2-s − 3.35·3-s + 5.50·4-s + 10.3·6-s − 7·7-s − 4.65·8-s + 2.27·9-s − 18.4·12-s − 11.9·13-s + 21.5·14-s − 7.68·16-s − 22.0·17-s − 7.01·18-s + 0.933·19-s + 23.5·21-s − 42.2·23-s + 15.6·24-s + 25·25-s + 36.8·26-s + 22.5·27-s − 38.5·28-s + 42.3·32-s + 67.9·34-s + 12.5·36-s + 37.3·37-s − 2.87·38-s + 40.1·39-s + ⋯ |
| L(s) = 1 | − 1.54·2-s − 1.11·3-s + 1.37·4-s + 1.72·6-s − 7-s − 0.581·8-s + 0.252·9-s − 1.54·12-s − 0.918·13-s + 1.54·14-s − 0.480·16-s − 1.29·17-s − 0.389·18-s + 0.0491·19-s + 1.11·21-s − 1.83·23-s + 0.650·24-s + 25-s + 1.41·26-s + 0.836·27-s − 1.37·28-s + 1.32·32-s + 1.99·34-s + 0.348·36-s + 1.01·37-s − 0.0757·38-s + 1.02·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2174793486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2174793486\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 41 | \( 1 - 41T \) |
| good | 2 | \( 1 + 3.08T + 4T^{2} \) |
| 3 | \( 1 + 3.35T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 11.9T + 169T^{2} \) |
| 17 | \( 1 + 22.0T + 289T^{2} \) |
| 19 | \( 1 - 0.933T + 361T^{2} \) |
| 23 | \( 1 + 42.2T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 37.3T + 1.36e3T^{2} \) |
| 43 | \( 1 - 85.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 85.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 108.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 143.T + 9.40e3T^{2} \) |
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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
−11.27886172447609143975246384340, −10.52757799904764787899276230483, −9.738446666584928970914783318207, −8.967250657618076172287333898068, −7.77387788726605826026593445359, −6.74085519066353334037442790990, −6.03334731166859995292416868010, −4.51556712705199215339340489020, −2.44049277061994159604840064260, −0.48052501210901266852092307548,
0.48052501210901266852092307548, 2.44049277061994159604840064260, 4.51556712705199215339340489020, 6.03334731166859995292416868010, 6.74085519066353334037442790990, 7.77387788726605826026593445359, 8.967250657618076172287333898068, 9.738446666584928970914783318207, 10.52757799904764787899276230483, 11.27886172447609143975246384340
