| L(s) = 1 | − 2·5-s + 2·13-s − 2·17-s − 19-s − 25-s − 2·29-s − 4·31-s + 2·37-s − 6·41-s − 4·43-s − 7·49-s − 10·53-s + 4·59-s − 2·61-s − 4·65-s − 12·67-s − 6·73-s − 4·79-s + 8·83-s + 4·85-s − 6·89-s + 2·95-s − 14·97-s + 6·101-s + 4·103-s + 20·107-s − 6·109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.554·13-s − 0.485·17-s − 0.229·19-s − 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.328·37-s − 0.937·41-s − 0.609·43-s − 49-s − 1.37·53-s + 0.520·59-s − 0.256·61-s − 0.496·65-s − 1.46·67-s − 0.702·73-s − 0.450·79-s + 0.878·83-s + 0.433·85-s − 0.635·89-s + 0.205·95-s − 1.42·97-s + 0.597·101-s + 0.394·103-s + 1.93·107-s − 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−9.083808553955238197086042174633, −8.333870839519268312655099491004, −7.64960308956647520776785924890, −6.78970818884896377763497497017, −5.92493690748268419984830773053, −4.82951180467259225179247200421, −3.97200836002451743759527577412, −3.13847923636082550280225437895, −1.70495479258263852028103331510, 0,
1.70495479258263852028103331510, 3.13847923636082550280225437895, 3.97200836002451743759527577412, 4.82951180467259225179247200421, 5.92493690748268419984830773053, 6.78970818884896377763497497017, 7.64960308956647520776785924890, 8.333870839519268312655099491004, 9.083808553955238197086042174633
