| L(s) = 1 | − 3-s + 2·5-s + 9-s + 4·11-s − 6·13-s − 2·15-s − 2·17-s − 4·19-s − 8·23-s − 25-s − 27-s − 2·29-s − 4·33-s − 10·37-s + 6·39-s + 6·41-s + 4·43-s + 2·45-s + 2·51-s + 6·53-s + 8·55-s + 4·57-s + 4·59-s − 6·61-s − 12·65-s − 4·67-s + 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 0.516·15-s − 0.485·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s − 1.64·37-s + 0.960·39-s + 0.937·41-s + 0.609·43-s + 0.298·45-s + 0.280·51-s + 0.824·53-s + 1.07·55-s + 0.529·57-s + 0.520·59-s − 0.768·61-s − 1.48·65-s − 0.488·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.825241684599749435471219697260, −7.67454257840429899018734503053, −6.92538722224681151756066962499, −6.18874160468740625251832036222, −5.60435712531698215962545180754, −4.60067949543777570406002885677, −3.91638973375108536412463827494, −2.39883079209175985717649788244, −1.69840313204406173914058397816, 0,
1.69840313204406173914058397816, 2.39883079209175985717649788244, 3.91638973375108536412463827494, 4.60067949543777570406002885677, 5.60435712531698215962545180754, 6.18874160468740625251832036222, 6.92538722224681151756066962499, 7.67454257840429899018734503053, 8.825241684599749435471219697260
