| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s + 2·13-s + 2·15-s + 6·17-s + 4·19-s + 21-s − 4·23-s − 25-s − 27-s − 6·29-s − 8·31-s + 2·35-s + 10·37-s − 2·39-s − 10·41-s − 12·43-s − 2·45-s − 8·47-s + 49-s − 6·51-s − 6·53-s − 4·57-s − 4·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s + 1.45·17-s + 0.917·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.338·35-s + 1.64·37-s − 0.320·39-s − 1.56·41-s − 1.82·43-s − 0.298·45-s − 1.16·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−9.414711671411861497929471048079, −8.119780722710108681675065767353, −7.69815802649856714297018450365, −6.75013471248181070621653252773, −5.81577366350407845158491505155, −5.09863361085506946239439169453, −3.83525472099276811420858805312, −3.32692818161216619186369980176, −1.52097630876353593040517597714, 0,
1.52097630876353593040517597714, 3.32692818161216619186369980176, 3.83525472099276811420858805312, 5.09863361085506946239439169453, 5.81577366350407845158491505155, 6.75013471248181070621653252773, 7.69815802649856714297018450365, 8.119780722710108681675065767353, 9.414711671411861497929471048079
