| L(s) = 1 | − 4.22·2-s + 8.50·3-s + 9.87·4-s − 5·5-s − 35.9·6-s + 15.5·7-s − 7.92·8-s + 45.3·9-s + 21.1·10-s + 16.8·11-s + 83.9·12-s − 13·13-s − 65.7·14-s − 42.5·15-s − 45.4·16-s + 23.6·17-s − 191.·18-s − 70.6·19-s − 49.3·20-s + 132.·21-s − 71.2·22-s + 35.4·23-s − 67.3·24-s + 25·25-s + 54.9·26-s + 155.·27-s + 153.·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.63·3-s + 1.23·4-s − 0.447·5-s − 2.44·6-s + 0.840·7-s − 0.350·8-s + 1.67·9-s + 0.668·10-s + 0.462·11-s + 2.01·12-s − 0.277·13-s − 1.25·14-s − 0.731·15-s − 0.710·16-s + 0.337·17-s − 2.50·18-s − 0.852·19-s − 0.551·20-s + 1.37·21-s − 0.690·22-s + 0.321·23-s − 0.572·24-s + 0.200·25-s + 0.414·26-s + 1.11·27-s + 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 + 4.22T + 8T^{2} \) |
| 3 | \( 1 - 8.50T + 27T^{2} \) |
| 7 | \( 1 - 15.5T + 343T^{2} \) |
| 11 | \( 1 - 16.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 23.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 70.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 35.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 285.T + 2.43e4T^{2} \) |
| 37 | \( 1 + 50.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 228.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 420.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 554.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 162.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 4.85T + 2.05e5T^{2} \) |
| 61 | \( 1 + 180.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 414.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 172.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 180.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 974.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 555.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 105.T + 9.12e5T^{2} \) |
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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
−8.467913829710126287806788672761, −7.933948034964519279271831658021, −7.37843994602097213242062996588, −6.61092962669697086315571805000, −4.98409573980624943730699272345, −4.03723473410670629333565820204, −3.13153450109243405606322113549, −1.96425639173264786392580844587, −1.49750962847389792902876129662, 0,
1.49750962847389792902876129662, 1.96425639173264786392580844587, 3.13153450109243405606322113549, 4.03723473410670629333565820204, 4.98409573980624943730699272345, 6.61092962669697086315571805000, 7.37843994602097213242062996588, 7.933948034964519279271831658021, 8.467913829710126287806788672761
