| L(s) = 1 | + 2-s + 0.648·3-s + 4-s + 5-s + 0.648·6-s + 2.28·7-s + 8-s − 2.57·9-s + 10-s − 2.02·11-s + 0.648·12-s − 0.331·13-s + 2.28·14-s + 0.648·15-s + 16-s + 4.82·17-s − 2.57·18-s + 1.78·19-s + 20-s + 1.48·21-s − 2.02·22-s − 0.804·23-s + 0.648·24-s + 25-s − 0.331·26-s − 3.61·27-s + 2.28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.374·3-s + 0.5·4-s + 0.447·5-s + 0.264·6-s + 0.862·7-s + 0.353·8-s − 0.859·9-s + 0.316·10-s − 0.611·11-s + 0.187·12-s − 0.0920·13-s + 0.609·14-s + 0.167·15-s + 0.250·16-s + 1.17·17-s − 0.607·18-s + 0.409·19-s + 0.223·20-s + 0.323·21-s − 0.432·22-s − 0.167·23-s + 0.132·24-s + 0.200·25-s − 0.0650·26-s − 0.696·27-s + 0.431·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.576071233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.576071233\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 - 0.648T + 3T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 + 2.02T + 11T^{2} \) |
| 13 | \( 1 + 0.331T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 1.78T + 19T^{2} \) |
| 23 | \( 1 + 0.804T + 23T^{2} \) |
| 29 | \( 1 + 4.48T + 29T^{2} \) |
| 37 | \( 1 - 7.59T + 37T^{2} \) |
| 41 | \( 1 - 6.39T + 41T^{2} \) |
| 43 | \( 1 + 3.68T + 43T^{2} \) |
| 47 | \( 1 - 4.32T + 47T^{2} \) |
| 53 | \( 1 - 5.16T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 3.82T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 2.21T + 71T^{2} \) |
| 73 | \( 1 - 2.52T + 73T^{2} \) |
| 79 | \( 1 + 3.73T + 79T^{2} \) |
| 83 | \( 1 - 0.932T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 5.37T + 97T^{2} \) |
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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
−7.80926527384127739818010945745, −7.04600789480001651288319291896, −6.05535854706660892562689490670, −5.48799902783416164884395271891, −5.15142503493575056952035274651, −4.17185361588902754878857897270, −3.40334307331207072806261043623, −2.61872200335748288061085664511, −2.03072927460009895544116112077, −0.909275180967742070968592821220,
0.909275180967742070968592821220, 2.03072927460009895544116112077, 2.61872200335748288061085664511, 3.40334307331207072806261043623, 4.17185361588902754878857897270, 5.15142503493575056952035274651, 5.48799902783416164884395271891, 6.05535854706660892562689490670, 7.04600789480001651288319291896, 7.80926527384127739818010945745
