| L(s) = 1 | + 3-s − 5-s − 2·7-s − 2·9-s + 3·11-s + 13-s − 15-s − 2·21-s − 4·23-s − 4·25-s − 5·27-s + 29-s − 3·31-s + 3·33-s + 2·35-s + 8·37-s + 39-s − 6·41-s − 5·43-s + 2·45-s − 3·47-s − 3·49-s − 5·53-s − 3·55-s − 8·59-s + 4·63-s − 65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s − 2/3·9-s + 0.904·11-s + 0.277·13-s − 0.258·15-s − 0.436·21-s − 0.834·23-s − 4/5·25-s − 0.962·27-s + 0.185·29-s − 0.538·31-s + 0.522·33-s + 0.338·35-s + 1.31·37-s + 0.160·39-s − 0.937·41-s − 0.762·43-s + 0.298·45-s − 0.437·47-s − 3/7·49-s − 0.686·53-s − 0.404·55-s − 1.04·59-s + 0.503·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.843378112255067891603077440940, −8.137668234105484587481230954798, −7.40530733603051725388723328323, −6.36230490937615343495442172496, −5.86876727871575445549099462889, −4.52492151385688776853632464015, −3.63636611349573586681053352594, −3.02549170589439465302071941358, −1.74689488016395042854701773575, 0,
1.74689488016395042854701773575, 3.02549170589439465302071941358, 3.63636611349573586681053352594, 4.52492151385688776853632464015, 5.86876727871575445549099462889, 6.36230490937615343495442172496, 7.40530733603051725388723328323, 8.137668234105484587481230954798, 8.843378112255067891603077440940
