| L(s) = 1 | + 3.97·2-s + 3·3-s + 7.83·4-s + 10.6·5-s + 11.9·6-s − 21.4·7-s − 0.663·8-s + 9·9-s + 42.2·10-s + 11·11-s + 23.5·12-s + 35.8·13-s − 85.3·14-s + 31.8·15-s − 65.3·16-s − 122.·17-s + 35.8·18-s − 50.9·19-s + 83.2·20-s − 64.3·21-s + 43.7·22-s − 49.5·23-s − 1.98·24-s − 11.9·25-s + 142.·26-s + 27·27-s − 168.·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.577·3-s + 0.979·4-s + 0.950·5-s + 0.812·6-s − 1.15·7-s − 0.0293·8-s + 0.333·9-s + 1.33·10-s + 0.301·11-s + 0.565·12-s + 0.763·13-s − 1.62·14-s + 0.548·15-s − 1.02·16-s − 1.75·17-s + 0.468·18-s − 0.614·19-s + 0.930·20-s − 0.668·21-s + 0.424·22-s − 0.449·23-s − 0.0169·24-s − 0.0959·25-s + 1.07·26-s + 0.192·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 3.97T + 8T^{2} \) |
| 5 | \( 1 - 10.6T + 125T^{2} \) |
| 7 | \( 1 + 21.4T + 343T^{2} \) |
| 13 | \( 1 - 35.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 259.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 404.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 478.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 102.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 604.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 127.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 236.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 986.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 302.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 10.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 976.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 649.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.26e3T + 9.12e5T^{2} \) |
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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
−8.652353853266895695276766847323, −7.26847772674837623857225734852, −6.32437811134886128773719488073, −6.19361043330724161780036659188, −5.17056499945480851395057793286, −4.04649947585376368795195867551, −3.63959424540926959497648300149, −2.52983485972121303230387007684, −1.89796908297971833376715874178, 0,
1.89796908297971833376715874178, 2.52983485972121303230387007684, 3.63959424540926959497648300149, 4.04649947585376368795195867551, 5.17056499945480851395057793286, 6.19361043330724161780036659188, 6.32437811134886128773719488073, 7.26847772674837623857225734852, 8.652353853266895695276766847323
