| L(s) = 1 | + 4.07·2-s − 3·3-s + 8.57·4-s + 17.2·5-s − 12.2·6-s − 8.63·7-s + 2.35·8-s + 9·9-s + 70.2·10-s − 11·11-s − 25.7·12-s − 3.05·13-s − 35.1·14-s − 51.7·15-s − 59.0·16-s − 93.5·17-s + 36.6·18-s + 5.17·19-s + 148.·20-s + 25.9·21-s − 44.7·22-s + 61.2·23-s − 7.07·24-s + 172.·25-s − 12.4·26-s − 27·27-s − 74.1·28-s + ⋯ |
| L(s) = 1 | + 1.43·2-s − 0.577·3-s + 1.07·4-s + 1.54·5-s − 0.831·6-s − 0.466·7-s + 0.104·8-s + 0.333·9-s + 2.22·10-s − 0.301·11-s − 0.619·12-s − 0.0651·13-s − 0.671·14-s − 0.891·15-s − 0.922·16-s − 1.33·17-s + 0.479·18-s + 0.0625·19-s + 1.65·20-s + 0.269·21-s − 0.434·22-s + 0.555·23-s − 0.0601·24-s + 1.38·25-s − 0.0938·26-s − 0.192·27-s − 0.500·28-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 | \( 1 - 4.07T + 8T^{2} \) |
| 5 | \( 1 - 17.2T + 125T^{2} \) |
| 7 | \( 1 + 8.63T + 343T^{2} \) |
| 13 | \( 1 + 3.05T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.17T + 6.85e3T^{2} \) |
| 23 | \( 1 - 61.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 109.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 378.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 244.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 7.20T + 1.03e5T^{2} \) |
| 53 | \( 1 + 652.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 173.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 205.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 813.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 243.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 378.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 479.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 314.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.03e3T + 9.12e5T^{2} \) |
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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
−8.518985448524148884156513303274, −6.97650959763361425533955609245, −6.51646719182893958989292351163, −5.90535014724490226077974499538, −5.13800941332271474906769621845, −4.65295454069671223034191207511, −3.44035805674771958585820991076, −2.53753875341419879617784221296, −1.67327698287507289476012707824, 0,
1.67327698287507289476012707824, 2.53753875341419879617784221296, 3.44035805674771958585820991076, 4.65295454069671223034191207511, 5.13800941332271474906769621845, 5.90535014724490226077974499538, 6.51646719182893958989292351163, 6.97650959763361425533955609245, 8.518985448524148884156513303274
