| L(s) = 1 | + 0.517·2-s − 1.73·4-s + 2.44·7-s − 1.93·8-s + 1.26·11-s + 1.79·13-s + 1.26·14-s + 2.46·16-s + 1.41·17-s − 3.26·19-s + 0.656·22-s − 6.31·23-s + 0.928·26-s − 4.24·28-s − 29-s − 8.73·31-s + 5.13·32-s + 0.732·34-s − 9.14·37-s − 1.69·38-s + 6.92·41-s − 9.14·43-s − 2.19·44-s − 3.26·46-s + 1.41·47-s − 1.00·49-s − 3.10·52-s + ⋯ |
| L(s) = 1 | + 0.366·2-s − 0.866·4-s + 0.925·7-s − 0.683·8-s + 0.382·11-s + 0.497·13-s + 0.338·14-s + 0.616·16-s + 0.342·17-s − 0.749·19-s + 0.139·22-s − 1.31·23-s + 0.182·26-s − 0.801·28-s − 0.185·29-s − 1.56·31-s + 0.908·32-s + 0.125·34-s − 1.50·37-s − 0.274·38-s + 1.08·41-s − 1.39·43-s − 0.331·44-s − 0.481·46-s + 0.206·47-s − 0.142·49-s − 0.430·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.517T + 2T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 - 1.79T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 + 6.31T + 23T^{2} \) |
| 31 | \( 1 + 8.73T + 31T^{2} \) |
| 37 | \( 1 + 9.14T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 9.14T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 - 5.93T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 7.34T + 73T^{2} \) |
| 79 | \( 1 + 4.19T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.80789758412847980233189520334, −6.93425240330332857408487083061, −6.03067613461410320420102813341, −5.44192981413813156101011503834, −4.79516318428975844849793374349, −3.91466585463501622693075913096, −3.60684778416512167519656887001, −2.22519095817124461056364015185, −1.35060707494952085711132267385, 0,
1.35060707494952085711132267385, 2.22519095817124461056364015185, 3.60684778416512167519656887001, 3.91466585463501622693075913096, 4.79516318428975844849793374349, 5.44192981413813156101011503834, 6.03067613461410320420102813341, 6.93425240330332857408487083061, 7.80789758412847980233189520334
