| L(s) = 1 | + 3-s − 2·4-s + 9-s − 2·11-s − 2·12-s − 3·13-s + 4·16-s + 2·17-s + 3·19-s + 23-s − 5·25-s + 27-s + 6·29-s − 3·31-s − 2·33-s − 2·36-s − 3·37-s − 3·39-s + 3·43-s + 4·44-s − 10·47-s + 4·48-s + 2·51-s + 6·52-s − 4·53-s + 3·57-s − 6·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1/3·9-s − 0.603·11-s − 0.577·12-s − 0.832·13-s + 16-s + 0.485·17-s + 0.688·19-s + 0.208·23-s − 25-s + 0.192·27-s + 1.11·29-s − 0.538·31-s − 0.348·33-s − 1/3·36-s − 0.493·37-s − 0.480·39-s + 0.457·43-s + 0.603·44-s − 1.45·47-s + 0.577·48-s + 0.280·51-s + 0.832·52-s − 0.549·53-s + 0.397·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.139724579032720143417745504450, −7.78198452576457114093017906499, −6.93331412463832529840576608792, −5.77854950504376604927628014169, −5.08581469342445398531678332691, −4.40757865375742754939843741156, −3.45408319458734500515786163355, −2.72242920681807393543289363904, −1.43333689923234733024938417597, 0,
1.43333689923234733024938417597, 2.72242920681807393543289363904, 3.45408319458734500515786163355, 4.40757865375742754939843741156, 5.08581469342445398531678332691, 5.77854950504376604927628014169, 6.93331412463832529840576608792, 7.78198452576457114093017906499, 8.139724579032720143417745504450
