| L(s) = 1 | − 2.62·2-s + 3-s + 4.91·4-s + 0.484·5-s − 2.62·6-s + 0.944·7-s − 7.67·8-s + 9-s − 1.27·10-s − 2.81·11-s + 4.91·12-s + 7.16·13-s − 2.48·14-s + 0.484·15-s + 10.3·16-s + 17-s − 2.62·18-s − 1.71·19-s + 2.38·20-s + 0.944·21-s + 7.40·22-s − 8.53·23-s − 7.67·24-s − 4.76·25-s − 18.8·26-s + 27-s + 4.64·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.577·3-s + 2.45·4-s + 0.216·5-s − 1.07·6-s + 0.356·7-s − 2.71·8-s + 0.333·9-s − 0.402·10-s − 0.848·11-s + 1.41·12-s + 1.98·13-s − 0.663·14-s + 0.125·15-s + 2.58·16-s + 0.242·17-s − 0.619·18-s − 0.394·19-s + 0.532·20-s + 0.206·21-s + 1.57·22-s − 1.77·23-s − 1.56·24-s − 0.953·25-s − 3.69·26-s + 0.192·27-s + 0.877·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 5 | \( 1 - 0.484T + 5T^{2} \) |
| 7 | \( 1 - 0.944T + 7T^{2} \) |
| 11 | \( 1 + 2.81T + 11T^{2} \) |
| 13 | \( 1 - 7.16T + 13T^{2} \) |
| 19 | \( 1 + 1.71T + 19T^{2} \) |
| 23 | \( 1 + 8.53T + 23T^{2} \) |
| 29 | \( 1 - 2.72T + 29T^{2} \) |
| 31 | \( 1 - 0.0695T + 31T^{2} \) |
| 37 | \( 1 + 8.64T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 2.32T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 3.05T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 5.34T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 5.43T + 73T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 3.31T + 97T^{2} \) |
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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
−8.157723129049937283790393395031, −7.933144154507360130940751654805, −6.78370917400185080260178112166, −6.25895599656190163985749762920, −5.38340414054355538640612930584, −3.92081462566648322496664636483, −3.08490132670969948795496397677, −1.94745232369497896109122384087, −1.49435936607259625376108300103, 0,
1.49435936607259625376108300103, 1.94745232369497896109122384087, 3.08490132670969948795496397677, 3.92081462566648322496664636483, 5.38340414054355538640612930584, 6.25895599656190163985749762920, 6.78370917400185080260178112166, 7.933144154507360130940751654805, 8.157723129049937283790393395031
