| L(s) = 1 | + 1.57·2-s + 3-s + 0.480·4-s + 0.664·5-s + 1.57·6-s − 4.15·7-s − 2.39·8-s + 9-s + 1.04·10-s − 11-s + 0.480·12-s + 2.20·13-s − 6.53·14-s + 0.664·15-s − 4.73·16-s − 0.555·17-s + 1.57·18-s − 6.78·19-s + 0.319·20-s − 4.15·21-s − 1.57·22-s − 4.88·23-s − 2.39·24-s − 4.55·25-s + 3.47·26-s + 27-s − 1.99·28-s + ⋯ |
| L(s) = 1 | + 1.11·2-s + 0.577·3-s + 0.240·4-s + 0.297·5-s + 0.643·6-s − 1.56·7-s − 0.846·8-s + 0.333·9-s + 0.331·10-s − 0.301·11-s + 0.138·12-s + 0.612·13-s − 1.74·14-s + 0.171·15-s − 1.18·16-s − 0.134·17-s + 0.371·18-s − 1.55·19-s + 0.0714·20-s − 0.906·21-s − 0.335·22-s − 1.01·23-s − 0.488·24-s − 0.911·25-s + 0.681·26-s + 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.57T + 2T^{2} \) |
| 5 | \( 1 - 0.664T + 5T^{2} \) |
| 7 | \( 1 + 4.15T + 7T^{2} \) |
| 13 | \( 1 - 2.20T + 13T^{2} \) |
| 17 | \( 1 + 0.555T + 17T^{2} \) |
| 19 | \( 1 + 6.78T + 19T^{2} \) |
| 23 | \( 1 + 4.88T + 23T^{2} \) |
| 29 | \( 1 - 6.82T + 29T^{2} \) |
| 31 | \( 1 + 4.34T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 + 2.62T + 41T^{2} \) |
| 43 | \( 1 - 7.39T + 43T^{2} \) |
| 47 | \( 1 - 8.12T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 67 | \( 1 + 0.106T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 3.84T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 8.42T + 97T^{2} \) |
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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
−8.864395615356647678355792220391, −8.058229933136402690461436461695, −6.81756458118293595693216014783, −6.23799866378358106098280434340, −5.68081783203300738813434981973, −4.42127032148814119000419442665, −3.79780924159023578884274942613, −3.03203116560229591975168470441, −2.14642983136790880957693493631, 0,
2.14642983136790880957693493631, 3.03203116560229591975168470441, 3.79780924159023578884274942613, 4.42127032148814119000419442665, 5.68081783203300738813434981973, 6.23799866378358106098280434340, 6.81756458118293595693216014783, 8.058229933136402690461436461695, 8.864395615356647678355792220391
